Question:
### SubjectName: Linear Equations ### ConstructName: Solve two-step linear equations, with the variable on one side, with all positive integers ### Question: Tom and Katie are arguing about a first step to solve this equation:
\(
\frac{m}{4}-1=20
\)
Tom says you can write: \( \frac{m}{4}=21 \)
Katie says you can write: \( m-1=80 \)
Who is correct?

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

Confuses the direction of vectors when adding or subtracting

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

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

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

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

Ignores the negative signs when adding positive and negative numbers

When adding column vectors, multiplies rather than adds

Forgets to add starting value when calculating new coordinates

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

Thinks the movement of the top number in a column vector is vertical, and the bottom number is horizontal rather than the other way round

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

Does not recognise the origin as the starting point for coordinates

Thinks a positive y value in a column vector means to move down rather than up

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

Thinks a negative y value in a column vector means to move up rather than down

Heads towards zero when subtracting from a negative

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

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

Believes that vectors of different magnitudes can be equal.

Believes adding two negatives gives a positive answer

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

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

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

Does not know the y intercept is where the x coordinate is 0

Doesn't recognise commutativity of addition with negative numbers

Believes that the y intercept is where y = 0

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

Assumes that an arrow on a number line starts at zero

Believes the centre of rotation is always the origin

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

Does not know that 7 and -7 are different

Does not realise addition is commutative

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

Believes the graph of x squared + c has the turning point of (-c, 0)

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

Believes we move down a number line when we add

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

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

Reflects in an axis, instead of the given mirror line

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: Perimeter ### ConstructName: Given the area of a square, calculate a missing dimension ### Question: A shape is made up a square and regular pentagon.
The area of the square is \( 144 \mathrm{~cm}^{2} \).

What is the perimeter of the compound shape? ![Pentagon and square joined along one edge]()

Adds when calculating area instead of multiplying

Multiplies up without finding unit price in proportion question

Finds unit price then forgets to multiply up

Does not divide to get a unitary amount before scaling up

Finds unit price but doesn't multiply up for number in question

Subtracts rather than dividing to find unit price

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

Does not know how to calculate with exchange rates

Assumes a 1:1 conversion ratio between different currencies

Divides when converting to a larger unit

Multiplies when converting to a larger unit

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

Multiplies up without finding unit amount

In a worded problem, does not complete all calculations

Believes they can round when converting units

Does not recognise inches (") as a unit of length

When reading values from graphs, writes down the wrong units

Believes direct proportion means a 1:1 ratio

Forgets to multiply up for the correct time frame in a worded question

Multiplies rather than divides when answering worded problems

Finds unit amount but doesn't multiply up

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

Believes they always divide by the conversion factor when converting units

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

Divides when converting to a smaller unit

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

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

Ends up £1 out when dealing with change in money

Rounds to the wrong degree of accuracy (rounds too little)

Rounds the wrong way when interpreting a real-life context

When converting compound measures into different units, thinks you multiply when the unit of the divisor has become smaller

Cannot spot a coefficient within a written scenario

Does not recognise litres as a unit of volume

Believes degrees can be used to measure length

Rounds to the wrong degree of accuracy (rounds too much)

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

Answers as if there are 100 minutes in an hour

Multiplies the value as if the original recipe was for 1 person

Only includes full hours when measuring time in minutes

Difficulty with translating a mathematical answer into a real world context

Multiplies instead of divides to find one part given the total

Does not recognise feet (ft) as a unit of length

Divides rather than multiplies when answering worded problems

Believes you can go above an amount of money when finding how much of something you can buy

Does not consider fractional parts when solving proportion problems

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 brackets ### Question: \( 4+12 \div(2+2)= \)

Confuses reciprocal and quadratic graphs

Confuses quadratic and exponential graphs

Only recognises the negative form of a quadratic graph

Believes a reciprocal equation will have a quadratic graph

Only recognises the positive form of a quadratic graph

Confuses reciprocal and exponential graphs

Believes that a positive x squared coefficient in a quadratic equation results in an n shaped graph

Believes a cubic equation will have a quadratic graph

Confuses the graph of a linear equation with a quadratic equation

Believes both solutions of a quadratic will have the same absolute value, even when the graph is not symmetrical about the y axis

Believes a reciprocal graph comes from a cubic equation

Does not recognise a reciprocal equation

Believes a quadratic graph can be a straight line

Believes you can square root a negative number

Thinks a decreasing sequence cannot be quadratic

Does not think a square root can be negative

Only gives 1 value of x for a given value of y, when reading from a parabolic graph

Selects y>a solutions instead of y<a solutions for a quadratic inequality

Believes a reciprocal equation will have a linear graph

Believes a cyclic quadrilateral can be self-intersecting

Believes that graphs of inverse proportion meet the axes

Believes the solution of a quadratic is its turning point

Believes that reciprocals are undefined for negative numbers

Believes a direct proportion graph can have a negative gradient

Believes multiplying two negatives gives a negative answer

Believes a cubic equation will have a linear graph

When factorising a quadratic without a non variable term, tries to double bracket factorise

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

Confuses the graph of 1/x with 1/x² when both are shown

When sketching graphs of factorised quadratics, does not see that the negative of the non variable terms in the brackets give the x intercepts or the roots of the graph.

Believes a polygon can have curved sides

Believes a reciprocal graph can be a straight line

Does not realise that a curved line represents a changing gradient

Believes an upward line has a negative gradient

Confuses the factorised form of repeated roots and the difference of 2 squares

Confuses the graphs of  direct and inverse proportion

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

Has an incorrect assumption about the definition of a reciprocal

Does not recognise factorising a quadratic

Believes the solutions of a quadratic equation are the constants in the factorised form

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: Recognise the place value of each digit in integers ### Question: What is the value of the \( 3 \) in the number below?
\(
43075
\)

Confuses the ten thousands and thousands place value columns

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

Writes the variable before the coefficient in an algebraic term

When collecting like terms, assigns positivity or negativity incorrectly.

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

Does not understand when like terms can be collected

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

Does not know that a single letter labels a vertex

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

Ignores coefficients when forming an expression from a question

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

Confuses collecting like terms with multiplication

Does not realize we can simplify a term that involves powers

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

Does not realise that two letters together are multiplying

Writes repeated letters instead of using index form of a variable

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

Cannot spot a variable within a written scenario

Believes a variable must remain unknown and cannot be substituted with a value

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

Thinks you can ignore the variable when simplifying algebraic fractions

Ignores the negative sign when thinking about the value of a term

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

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

Does not understand how to multiply algebraic terms

Does not know the meaning of the identity symbol

Believes an algebraic term with the same coefficient as the value of a constant term can be subtracted to give zero

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

When simplifying, thinks they write 0 in front of a negative number in the final answer, instead of just writing the negative number

Does not understand use of the counters to represent algebra

Thinks that we can remove a division symbol and just write the variables next to each other

Believes the variable should remain in place after substituting its value

Believes that 'zero' is read as the letter 'O'

Ignores numerical terms when setting up an equation from a question

Does not know how to identify common factors from algebraic terms

Does not know the correct notation for labelling a side length

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

Does not know how to simplify algebraic fractions

Has incorrect belief about the meaning of "expression"

Multiplies all terms together when simplifying an expression

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: Difference of Two Squares ### ConstructName: Factorise a quadratic expression in the form x² - c ### Question: Which of the following answers shows a correct factorisation of the expression below? \( x^{2}-9 \)

Does not recognise difference of two squares

Assumes the item has not been replaced in a probability question where it has actually been replaced

Does not reduce the denominator when an item is not replaced in a probability question

Thinks the denominator in a combined probablity will be the same as the denominator in the probability of the first event.

Does not recognise that choosing 2 items at once means the first item does not get replaced in a probability question

Thinks the order of events does not matter in a question about combined probability with no replacement

Believes the numerator stays the same when calculating a second pick of the same type without replacement on a tree diagram

Believes the probability on all branches travelling in the same direction on a tree diagram are the same, despite changing events

Does not increase the denominator when an item is added in a probability question

Thinks they can just give the probability of the first event for a combined probability

Believes you always reduce the numerator and denominator by 1 when writing the probability for a second pick without replacement on a tree diagram

Believes when calculating a probability the denominator will be the total of one of the criteria rather than the overall total that is required (gives a given that answer)

Uses outcome as numerator in probability

Adds rather than multiplies for combined probabilities

Believes 'given that a, what is the probability of b?' means 'given that b, what is the probability of a?'

Gives the probability out of the number of non-successes rather than the total

Does not work out all possibilites when given an 'at least' question in probability

Adds the numerators when working out a combined probability

Believes any sample gives an equal chance of selection

Believes data implies equal chance of selection

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

When using the capture-recapture method, believes the number caught second is the numerator and the number marked second is the denominator

Includes too many possibilities when calculating a combined event in probability

Writes the numerator and the denominator the wrong way round in a probability

Confuses theoretical and experimental probability

Does not understand that the probability of something not happening is 1 minus the probability of it happening

Confuses dependent and mutually exclusive events

Believes all possible outcomes must have equal probability

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

Believes P(success) in probability notation means the number of successes

Believes a sample space should show repeated outcomes, not just unique possible outcomes.

Includes too many possibilites when given an 'at least' question in probability

Believes biased selection methods can be random

Does not understand the effect of consecutive subtractions

Cannot identify mutually exclusive events

Misunderstands the capture-recapture method

Does not understand the effect of consecutive operations

Lists outcomes rather than counts how many outcomes

Confuses relative frequency and expected frequency

Has taken the percentage from the original

Carries out operations from right to left regardless of priority order

Carries out operations from left to right regardless of priority order

When using the capture-recapture method of sampling, believes you divide the number caught second by the number marked second to find the best estimate of population size

In repeated percentage change, believes the second change is only a percentage of the first change, without including the original

When using the capture-recapture method for sampling, believes you subtract the number marked second from the number caught second to find the best estimate of population size

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

Does not understand how to maintain equivalence in fractions

Confuses all possible outcomes with options available

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 linear inequalities in one variable where the unknown appears on boths sides of the equation ### Question: Solve
\[
3 n+2 \geq 4 n-5
\]

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

When ordering integers, orders from the digits at the end of the number instead of the digits at the start

Mixes up smallest and greatest when ordering positive integers

Thinks the more digits a number has the greater it is, regardless of place value

Mixes up greater than and less than symbols

Orders integers based on first digits without taking into account place value

Believes negative numbers with a larger absolute value are greater

Believes digits of a number are read/written from right to left

Assumes that a list is given in size order

Chosen the longest value rather than the greatest

Mixes up smallest and largest when ordering decimals

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

When ordering decimals, believes the decimal point is not relevant

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

Does not equate the value of a number with the place value column it is in

Mixes up smallest and largest when ordering fractions

Subtracts the smallest digit from the largest digit

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

When placing a negative number on a number line student assumes numbers with a greater absolute value are placed to the right.

Believes you subtract the smaller digit from the larger digit in column subtraction, regardless of position

Writes digits in the order of the pictorial representation rather than using place value columns

Counts on in the wrong direction on a number line

Counts in the wrong direction on a number line

Believes that the larger the dividend, the smaller the answer.

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

Does not understand the meaning of "value"

Confuses the order of the digits in stick multiplication

Mixes up shortest and longest when ordering times

Does not understand the value of zeros as placeholders

Does not understand which place value is the one that determines if a number rounds up or down

When ordering numbers in standard form just looks at the decimal without multiplying by a power of 10

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

Ignores zeros when giving the value of an integer

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

Confuses the hundreds and hundred thousands place value columns

Believes that the larger the divisor, the larger the answer.

Believes the smaller number is always subtracted from the larger, regardless of the order of the calculation

Believes that fractions with larger denominators are greater in size

Believes a number written in standard form is smaller than another number written in standard form if the number between 1 and 10 is smaller

Does not know how to compare decimal numbers

Confuses the hundreds and thousands place value columns

Ordered fractions by comparing denominators rather than using equivalent fractions.

Confuses the hundreds and tens place value columns

Confuses the ten thousands and thousands place value columns

Changes wrong place value column when rounding

Believes that numbers are read aloud digit by digit

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

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: Probability of Single Events ### ConstructName: Assign probabilities to single events using fractions ### Question: A box contains \( 10 \) coloured counters.

\( 3 \) of the counters are pink

\( 3 \) of the counters are blue

The rest of the counters are green

What is the probability of choosing a green counter?

When finding the area of a parallelogram finds base x perpendicular height but divides the answer by 2

When finding a missing side of a parallelogram given the area completes an extra step of dividing by 2

When finding the area of a parallelogram does not understand which lengths to use

Calculates perimeter when asked for area

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

Does not realise we can find a missing length if we know the base of a parallelogram and its area.

Has found the area of a triangle rather than an area of a parallelogram

Believes h in the formula for area of a trapezium stands for half the sum of the parallel sides

Adds when calculating area instead of multiplying

Calculates area when asked for perimeter

Calculates circumference when asked for area

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

Multiplies all given dimensions when calculating an area

Uses formula for area of a parallelogram when finding the area of a trapezium

Finds the circumference instead of the area

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

Thinks they double rather than halve when finding the area of a triangle

Halves base but still divides product by 2 when finding area of a triangle

Thinks halving is the same as multiplying by 2

Thinks you multiply parallel sides to find the area of a trapezium

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

Does not half sum of parallel sides when finding area of a trapezium

adds all given dimensions when calculating area

Adds instead of subtracts when using an area model for multiplication

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

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

Uses dividing fractions method for multiplying fractions

Thinks all dimensions given must be used when calculating the area

Confuses formulae for area of triangle and area of rectangle

Mixes up width and height of a rectangle

Counts half-squares as full squares when calculating area on a square grid

Uses only the base and perpendicular height when calculating the area of a trapezium

Thinks measures of area must end in squared

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

Adds when calculating volume instead of multiplying

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

Thinks the inverse of subtraction is division

Does not have when calculating the area of a triangle

Thinks the inverse of addition is division

Thinks the inverse of subtraction is multiplication

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: Reflection ### ConstructName: Reflect an object or a point in a vertical line with the line of reflection given as an equation, where the line of reflection lies on the edge or outside the object ### Question: Which is the correct point for E when reflected in the line \( x=-1 \) ? ![A set of axes: x-axis from -5 to 5, y-axis from -5 to 3. Four points are plotted and joined to make a rectangle: (1,1), (3, 1), (3, 3) and (1,3). Point (3, 3) is labelled with the letter "E". ]()

Thinks x = ? is a horizontal line through ?

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

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

Does not know how to convert percentages with a decimal point into decimals

When converting a percentage to a decimal, just writes all the digits after a decimal point.

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

Divides instead of multiplies when converting a decimal to a percentage

Divides by 10 instead of 100 when converting a percentage to a decimal

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

Thinks you need to just remove the percentage sign to convert from a percentage to a decimal

Does not understand a percentage is out of 100

Does not know that you can have a decimal percentage

Multiplies by 10 instead of 100 when converting decimal to percentage

Assumes decimal point can be ignored when given a decimal percentage to convert to a fraction

Assumes that converting a decimal percentage to a fraction, they can just convert the decimal to a fraction, instead of writing the decimal over 100.

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

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

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

Divided by 100 rather than multiplied by 100

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

Believes a decimal with one significant figure is a multiple of 10 when converted to a percentage, regardless of place value

Believes percentages cannot be converted into fractions

Believes that a greater percentage represents a greater quantity

Converts a fraction to a decimal by using only the numerator after the decimal point

Multiplied by 100 rather than divided by 100

Converts fraction to percentage by putting % sign after the denominator

Thinks they just remove the decimal point when converting a decimal to a percentage

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

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

Error finding percentage value after finding 1%

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

Thinks that a percentage can be converted to a fraction using a denominator of 10

Converts fraction to percentage by multiplying the denominator by 10

Converts a fraction to a decimal by placing all values of the fraction after the decimal point

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

Believes they can reverse a percentage increase or decrease by dividing by the decimal multiplier of the original change

Does not understand changing decimals to fractions

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

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

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

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

Converts a fraction to a decimal by using the denominator after the decimal point

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

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

Found the percentage but forgot to add to 100%

Converts a fraction to a decimal by placing the denominator after the decimal point

Thinks they can convert percentages to fractions by writing the percentage value as both the numerator and 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: Squares, Cubes, etc ### ConstructName: Calculate the square root of a number ### Question: What is the value of the missing number? ![\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{\( 15^{2} \)} \\
\hline\( \square^{2} \) & \( 2 \) \\
\hline
\end{tabular}]()

Thinks the sum of 2 numbers, each squared first, is the same as adding the two numbers and then squaring the total

Adds instead of multiplying when expanding bracket

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

Ignores negative sign when expanding bracket

When expanding brackets, multiplies out the first term correctly but adds together the number outside the bracket with the second term inside the bracket.

Believes multiplying a positive by a negative gives a positive answer

Only multiplies the first term in the expansion of a bracket

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

Adds instead of multiplies when multiplying algebraic terms

Only multiplies second term in the expansion of a bracket

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

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

Believes they can ignore the coefficient when expanding double brackets

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

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

Ignores the negative signs when adding positive and negative numbers

Only multiplies the numerical terms when expanding a bracket

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

Believes adding two negatives gives a positive answer

Believes multiplying two negatives gives a negative answer

Believes they only need to multiply the first pair of terms when expanding a double bracket.

Believes that multiplying variables is the same as adding them together

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

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

Thinks the multiplication sign means to add

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

When factorising into double brackets, finds the correct values for the non variable terms but swops the plus and minus sign

Thinks two variables next to each other represent addition

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

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

Believes a cubic expression should have three terms

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

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

Believes cubing a number means you multiply it by 3

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

Believes 0 multiplied by a number gives the number

Believes you add the terms together when multiplying like terms.

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

Believes the solutions of a quadratic equation are the absolute values of the constants in the factorised form

When solving an equation, multiplies instead of adding

Thinks multiplication and addition are the same

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

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

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: Estimation ### ConstructName: Round numbers to an appropriate degree to estimate the answer to a calculation ### Question: When estimating the answer to a complicated problem involving \( 0.482 \), you should instead use...

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

Believes that fractions with larger denominators are greater in size

Ordered fractions by comparing denominators rather than using equivalent fractions.

Mixes up smallest and largest when ordering fractions

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

When ordering integers, orders from the digits at the end of the number instead of the digits at the start

Believes negative numbers with a larger absolute value are greater

Mixes up greater than and less than symbols

Mixes up smallest and greatest when ordering positive integers

Mixes up smallest and largest when ordering decimals

Believes that fractions with the same denominator are equal

Does not find a common denominator when comparing fractions

Believes a smaller number of shaded parts represents a smaller fraction

When ordering decimals, believes the decimal point is not relevant

Assumes that a list is given in size order

Believes that a fraction means dividing the denominator by the numerator

Believes that the larger the dividend, the smaller the answer.

Believes that the smaller a divisor is, the smaller the answer will be

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

When placing a negative number on a number line student assumes numbers with a greater absolute value are placed to the right.

Thinks a greater number of shaded parts represents a greater fraction

Believes -a is always smaller than a, ignoring the possibility that a is negative

Believes that the larger the divisor, the larger the answer.

Does not understand the 'less than' inequality notation

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

Counts on in the wrong direction on a number line

Thinks numbers greater than 1 cannot be converted into fractions

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

Thinks the more digits a number has the greater it is, regardless of place value

Counts in the wrong direction on a number line

Assumes the denominator is divided by the numerator

When ordering numbers in standard form just looks at the decimal without multiplying by a power of 10

Does not know how to compare decimal numbers

Believes unit fractions with a difference of 1 in the denominator have no other fractions between them

Does not understand equivalent fractions

Gives the smallest whole when asked for the smallest fraction shaded, presented visually

Believes the sides of an inequality can be switched without changing the direction of the sign

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

Does not realise you can use equivalent fractions to break fractions up into smaller divisions

Does not know the units column is before the decimal point

Student thinks that negative decimals behave the same as positive decimals when positioning a decimal between two consecutive negative integers, placing the decimal with a greater absolute value to the right.

Chosen the longest value rather than the greatest

Mixes up colder and hotter when comparing temperatures

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

Thinks subtracting a negative makes your answer smaller

Mixes up shortest and longest when ordering times

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 class of students were surveyed about how many siblings they have.

A selection of results are shown in the table. \begin{tabular}{|c|c|c|}
\hline Number of siblings & Frequency & Angle on a pie chart \\
\hline  0  & \( 7 \) & \(  \) \\
\hline 1  & \(  \) & \( 144 \degree \) \\
\hline 2  & \(  \) & \(  \) \\
\hline 3  & \( 4 \) & \( \bigstar \) \\
\hline 4 or more  & \(  \) & \(  \) \\
\hline  TOTAL  & 30  & \(  \) \\
\end{tabular} Jenny wants to draw a pie chart of the information.
What should replace the star?

Believes you find the angle for a pie chart sector by doing its frequency multiplied by total frequency

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

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

Multiplies by 10 instead of 100 when converting decimal to percentage

Thinks you need to just remove the percentage sign to convert from a percentage to a decimal

Does not know how to convert percentages with a decimal point into decimals

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

Does not know that you can have a decimal percentage

Divides instead of multiplies when converting a decimal to a percentage

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

Does not understand a percentage is out of 100

Thinks they just remove the decimal point when converting a decimal to a percentage

When converting a percentage to a decimal, just writes all the digits after a decimal point.

Assumes that converting a decimal percentage to a fraction, they can just convert the decimal to a fraction, instead of writing the decimal over 100.

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

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

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

Believes a decimal with one significant figure is a multiple of 10 when converted to a percentage, regardless of place value

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

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

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

Assumes decimal point can be ignored when given a decimal percentage to convert to a fraction

Found the percentage but forgot to add to 100%

Does not understand changing decimals to fractions

Divides by 10 instead of 100 when converting a percentage to a decimal

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

Converts fraction to percentage by putting % sign after the denominator

Only reads the percentage symbol as applying to the units digit

Error finding percentage value after finding 1%

Does not understand how to find percentages of an amount

Believes percentages cannot be converted into fractions

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

Multiplied by 100 rather than divided by 100

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

Divided by 100 rather than multiplied by 100

Adds "100" to end of written value rather than multiplying by 100

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

Does not understand the value of zeros as placeholders

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

Does not know the units column is before the decimal point

Converts a fraction to a decimal by using only the numerator after the decimal point

Does not know how to compare decimal numbers

Believes that when increasing by a percentage over 100, they do not need to add an extra 1 to the decimal multiplier for the original amount

Has considered the percentage rather than the percentage of the amount

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

Thinks they can convert percentages to fractions by writing the percentage value as both the numerator and the denominator

Converts fraction to percentage by multiplying the denominator by 10

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 Percentages ### ConstructName: Convert single digit integer percentages to fractions ### Question: \( 4 \% \) as a fraction is....

Does not understand a percentage is out of 100

Multiplies up without finding unit price in proportion question

Does not divide to get a unitary amount before scaling up

Finds unit price but doesn't multiply up for number in question

Finds unit price then forgets to multiply up

Finds unit amount but doesn't multiply up

Multiplies up without finding unit amount

Believes direct proportion means a 1:1 ratio

Subtracts rather than dividing to find unit price

Multiplies the value as if the original recipe was for 1 person

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

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

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

In direct and inverse proportion, forgets to find the value of k, and carries out calculations as though k were 1

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

Does not believe you can scale up a proportional statement

Cannot spot a coefficient within a written scenario

Adds rather than multiplies when scaling up in direct proportion problems

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

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

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

Shares equally instead of in given ratio

Thinks direct proportion rules apply to inverse proportion problems

Divides rather than multiplies when scaling up in direct proportion problems

Does not know the meaning of the word "product"

Does not know how to calculate with exchange rates

Forgets to multiply up for the correct time frame in a worded question

In ratio where total is given, treats 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

Assumes value of symbols in a pictogram without checking the key

Gives number of sections of pie chart rather than the amount it represents

In a worded problem, does not complete all calculations

Multiplies instead of divides to find one part given the total

Finds a scale factor but forgets to multiply the original value in proportion questions

Assumes a 1:1 conversion ratio between different currencies

Believes every direct or inverse proportion question uses the formula p=kq

Difficulty with translating a mathematical answer into a real world context

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

Does not understand the meaning of "value"

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

Divides rather than multiplies in a related calculation questions

Subtracts rather than divides when scaling down in direct proportion problems

Mixes up constant and coefficient in a worded substitution problem

Estimates shares of a ratio instead of calculating

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

When interpreting a pictogram, thinks each symbol stands for 1

Is not able to spot a proportional relationship pictorially unless they are laid out in groups

Does not realise ratio in worded problem can be simplified

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: Identify when it is appropriate to use a line graph ### Question: Which of the following situations would a line graph be the most appropriate to use?

Does not understand that a line graph shows a relationship between 2 variables

Does not understand that factors are divisors which split the number into equal groups

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

Does not realise that division can be broken down into factors

Believes splitting a multiplication using the distributive property is the same as splitting it into factors

Does not understand a denominator represents equal parts

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

Does not realise you can use equivalent fractions to break fractions up into smaller divisions

Does not know how to interpret a fraction worded problem

Does not realize we can write a division as a fraction

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

Does not understand equivalent fractions

Misunderstands the concept of common factors in algebra

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

Does not understand that the numerator and denominator of fractions represent groupings and have the same order of priority as brackets

Mixes up the terms for the parts of a fraction

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

Does not realise that multiplication can be broken down into factors

Does not know how to divide a fraction by an integer

Believes numbers are divisible by numbers which sum to their factors

Does not know how to solve linear equations involving fractions

Does not recognise difference of two squares

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

Does not understand parts must be equal in a fraction diagram

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

Does not understand how to simplify fractions

Unable to equate two divisions with a single division

Thinks numbers are divisible by all multiples of their factors

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

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

Does not know how to simplify algebraic fractions

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

Does not correctly apply the distributive property of multiplication

Does not understand the relationship between a ratio and fraction

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

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

Can not recognise all factor pairs of numbers

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

Believes factors of algebraic terms/expressions will only be numbers

Believes stratified samples should be evenly split

Does not know how to identify common factors from algebraic terms

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

Thinks you can divide terms by different factors when simplifying an algebraic 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: Basic Angle Facts (straight line, opposite, around a point, etc) ### ConstructName: Recognise the vertically opposite angles relationship ### Question: What is the relationship between these two angles? ![A pair of lines crossing to form an X shape. Two angles are marked, with one blank angle between them on each side.]()

Does not know the meaning of the word parallel

Believes you can have decimals in a simplified fraction

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

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

Gives a fraction when asked for a decimal

Rounds a decimal instead of using a fraction in division

Does not understand how to simplify fractions

Has used the decimal point to separate numerator and denominator

Believes when simplifying a fraction you can divide the numerator and denominator by different values

Converts a fraction to a decimal by placing all values of the fraction after the decimal point

Does not understand changing decimals to fractions

Believes clock times can be written with a decimal point rather than a colon

Does not recognise when simplifying a fraction may be helpful

Does not understand that coordinates can be decimals

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

Thinks you can multiply by a decimal to find a multiple of a number

When simplifying an algebraic fraction, only looks for number factors

Believes integers can have decimal places

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

Converts a fraction to a decimal by using only the numerator after the decimal point

When adding fractions, adds the numerators and denominators

Thinks that a unit fraction can be simplified further

Believes fraction to decimal conversion is done by writing the numerator as a decimal

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

Thinks you cannot multiply by a decimal to get equivalent fractions

Believes the decimal point button writes a fraction

Thinks the fraction button on a calculator converts fractions to decimals

Divides instead of multiplies when multiplying by a decimal

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

Does not understand equivalent fractions

Does not recognise when a fraction can be simplified

When multiplying fractions, multiplies both the numerator and denominator

Thinks you can divide the variables and the numerical values by different numbers when simplifying an algebraic fraction

Gives a rounded whole number instead of a decimal

Converts a fraction to a decimal by using the denominator after the decimal point

Converts a fraction to a decimal by placing the denominator after the decimal point

Does not know how to simplify algebraic fractions

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

When subtracting fractions, subtracts the numerators and denominators

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

Believes that a fraction means dividing the denominator by the numerator

Does not realise you can use equivalent fractions to break fractions up into smaller divisions

Does not know how to convert percentages with a decimal point into decimals

Assumes fractions with prime numerators cannot be simplified

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

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: Real Life Graphs ### ConstructName: Given a situation sketch an appropriate real life graph ### Question: Water is poured into a glass at a constant rate.

This graph shows how the depth \( (d) \) of water in a glass changes over time \( (t) \).

Which of the following glasses is this most likely to be the graph of? ![A depth-time graph which is a curve starting at (0, 0), with a steep gradient to begin with but then the gradient decreases and the graph levels off to almost horizontal.]()

Does not spot the difference between a gradual change of gradient and 2 distinct line segments with different constant gradients

Struggles to visualise nets of 3D shapes

Does not realise a triangular prism is a 3D shape

Names one of the faces of a 3D shape rather than the 3D shape name

Thinks all lines on a net will form edges in 3D

Does not know how to calculate the volume of a prism

Struggles to visualise 2D triangles in a cuboid

Struggles to draw 3D shapes on isometric paper

Struggles to visualise elevations of 3D shapes

Believes a line showing a constant rate of increase corresponds to a non-prism shape

Does not know that a solid is a 3D shape

Does not know the properties of a cylinder

Struggles to visualise how an object will look in 3D when given a plan view, front and side elevation

Believes parts of a 3D diagram that look straight class as edges

Does not have when calculating the area of a triangle

Does not know the meaning of uniform cross section

Believes edges of a 3D shape do not include edges of base

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

Thinks depth is marked on the cross section of a prism

Thinks cross section of a prism can be any face

Believes only straight lines count as edges on 3D shapes

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

Thinks cube numbers are triangular numbers

Tries to apply sohcahtoa to a non right angled triangle

Thinks that when drawing a plan view, you only include the top layer of the shape

Confuses scalene and right angle triangles

Think cross section of a prism is the biggest face

Gives 3-letter angle notation in the wrong order

In 3 letter angle notation, gives a wider angle that contains the shaded angle

Thinks depth always goes front to back in a diagram of a 3D shape

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

Chooses the wrong 3 letters to describe an angle

Gives the vertex instead of the 3-letter angle notation

Confuses scalene and equilateral triangles

Believes triangle numbers are formed differently to the correct way

Thinks the orientation of the side view does not matter when identifying the direction of view

Does not understand the 3-letter convention for labelling angles

Believes that when constructing triangles, all sides should be measured from the same point

Thinks the plan view is from the side of the shape

Thinks 3-letter angle notation has to use the nearest vertex on each side of the angle

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

Draws both angles at the same end of the line when constructing a triangle

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: Basic Calculator Use ### ConstructName: Perform calculator calculations with powers ### Question: Which button would you press to raise a number to a power? Which button would you press to raise a number to a power? ![A scientific calculator]() ![A scientific calculator]()

Thinks the fraction button on a calculator converts fractions to decimals

Believes the sides of a rectangle must be horizontal and vertical, not diagonal

Believes the sides of a square must be horizontal and vertical, not diagonal

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

Believes a rectangle has four equal sides

Does not know the properties of a rectangle

Makes an assumption about line segments being equal within a shape

Does not recall rectangle side length property

Believes a rectangle has one pair of equal sides

Mixes up width and height of a rectangle

Reflects in an axis, instead of the given mirror line

Uses the same value for both coordinates

Thinks x = ? is a horizontal line through ?

Does not realise that if the horizontal and vertical measurements of a diagonal line segment switch it remains the same size

Includes the x variable when giving the equation of a horizontal line

Does not know how to find the y intercept when given two other sets of coordinates that lie on a straight line

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

Believes a line segment can be opposite an axis

Believes that one point is enough to specify a line

Does not know that we can use the conservation of perimeter to find missing lengths

Does not know the meaning of the word parallel

Confuses lines of the form y = a and y= ax

Thinks y = ? is a vertical line through ?

Believes that points on an axis have equal x and y coordinates

Not considered the y value when reading coordinates

Forgets to add starting value when calculating new coordinates

Believes lines of reflection must be vertical or horizontal

Believes lines are parallel if they have the same y intercept

Believes two lines of the same length are parallel

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

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

Believes if a point is reflected in the line x=a then the x coordinate of the point will become 'a'

Believes the space diagonal is the same length as the diagonal on the bottom face of a cuboid

Mixes up the x and y values of the coordinates

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

Believes lines can be parallel by having the same y intercept OR gradient

Believes there is more than one line which passes through a given pair of points

Believes that the y intercept is (0, y) where y is any y coordinate on the line

Believes a horizontal line can show a constant rate of change

Does not know what dashes on the sides of shapes mean

Does not know the information each vertical line of a box plot represents

Thinks you can find missing values in a given table by treating the row as linear and adding on the difference between the first two values given.

Given a line segment with a midpoint that is not the origin, assumes that the two end points will still have the same absolute value

When a point is on an axis, gives both coordinates as the value it's at on the axis

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: Properties of Quadrilaterals ### ConstructName: Identify the order of rotational symmetry of a shape ### Question: Which shape has the lowest order of rotational symmetry?

Does not know how to find order of rotational symmetry

Mixes up smallest and largest when ordering fractions

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

Ordered fractions by comparing denominators rather than using equivalent fractions.

Believes that fractions with larger denominators are greater in size

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

When ordering integers, orders from the digits at the end of the number instead of the digits at the start

Does not find a common denominator when comparing fractions

Mixes up smallest and greatest when ordering positive integers

Mixes up smallest and largest when ordering decimals

Believes negative numbers with a larger absolute value are greater

Assumes that a list is given in size order

Believes that fractions with the same denominator are equal

Mixes up greater than and less than symbols

When ordering decimals, believes the decimal point is not relevant

Believes a smaller number of shaded parts represents a smaller fraction

Thinks a greater number of shaded parts represents a greater fraction

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

Does not understand equivalent fractions

Does not know how to compare decimal numbers

Believes that a fraction means dividing the denominator by the numerator

Chosen the longest value rather than the greatest

When placing a negative number on a number line student assumes numbers with a greater absolute value are placed to the right.

Does not realise you can use equivalent fractions to break fractions up into smaller divisions

Thinks numbers greater than 1 cannot be converted into fractions

Does not understand the 'less than' inequality notation

Believes that the larger the dividend, the smaller the answer.

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

Believes unit fractions with a difference of 1 in the denominator have no other fractions between them

Believes that the larger the divisor, the larger the answer.

Believes that the smaller a divisor is, the smaller the answer will be

Does not understand how to maintain equivalence in fractions

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

Believes -a is always smaller than a, ignoring the possibility that a is negative

Thinks the more digits a number has the greater it is, regardless of place value

Mixes up shortest and longest when ordering times

When ordering numbers in standard form just looks at the decimal without multiplying by a power of 10

Counts on in the wrong direction on a number line

Applies BIDMAS in strict order (does not realize addition and subtraction, and multiplication and division, are of equal priority)

Counts in the wrong direction on a number line

Does not understand that the numerator and denominator of fractions represent groupings and have the same order of priority as brackets

Uses numerators to convert fractions to ratio despite different denominators

Believes the smaller number is always subtracted from the larger, regardless of the order of the calculation

Gives the smallest whole when asked for the smallest fraction shaded, presented visually

Thinks that a larger shaded area represents a larger fraction

Forgets to change the numerators when finding equivalent fractions

Believes the sides of an inequality can be switched without changing the direction of 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: BIDMAS ### ConstructName: Use the order of operations to carry out calculations involving brackets ### Question: \[
9-8+5
\]
Jo and Paul are trying to make the calculation above equal to \( 6 \).
Jo says you can write it like this: \( 9-(8+5) \)
Paul says you can leave it like this: \( 9-8+5 \)
Who is correct?

Applies BIDMAS in strict order (does not realize addition and subtraction, and multiplication and division, are of equal priority)

Has added the powers rather than multiplying them

Multiplied the powers rather than added them

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

Adds instead of multiplies when multiplying algebraic terms

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

When multiplying variables, thinks they add coefficients as well as powers

Thinks the multiplication sign means to add

Adds instead of multiplying when expanding bracket

Believes that multiplying variables is the same as adding them together

Believes squaring is the same as adding two

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

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.

Thinks two variables next to each other represent addition

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

Thinks index notation represents repeated addition rather than repeated multiplication

Thinks multiplication and addition are the same

Mixes up squaring and multiplying by 2 or doubling

Multiplies coefficients and letters separately and writes as an addition

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

Believes you multiply matching bases when carrying out a multiplication

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

When multiplying fractions, adds the numerators and multiplies the denominators

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

When multiplying fractions, adds the numerators and the denominators

Added rather than multiplied by a scale factor

Adds rather than multiplies when asked for a product

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

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

Believes you add the terms together when multiplying like terms.

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

When multiplying numbers with the same base, multiplies the powers

Thinks the subtraction sign means multiply

Assumes they multiply by the numerator of fractional index

Believes addition of terms and powers of terms are equivalent e.g. a + c = a^c

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

Added the values together instead of finding the 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: Volume and Capacity Units ### ConstructName: Given the perimeter, work out a missing length requiring knowledge of the geometrical properties of the shape ### Question: A regular pentagon has a total perimeter of \( 17 \mathrm{~cm} \).
What is the length of each of its sides?

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

Believes a curve can show a constant rate

Believes a horizontal line can show a constant rate of change

Believes a line showing a constant rate of increase corresponds to a non-prism shape

Does not know the properties of a cylinder

Does not know the meaning of uniform cross section

Believes infinite gradient is not possible in real life.

Believes a direct proportion graph is a horizontal line

Does not realise that a curved line represents a changing gradient

Does not interpret a sloping gradient as a change in rate

Believes a steeper downwards line on a graph means a slower decrease

Believes all linear graphs show direct proportion

Believes a reciprocal equation will have a linear graph

Does not know the properties of a linear graph

Believes that a general trend of an increase or decrease over time, on a statistical diagram, means that the slope must be in the same direction for the whole time period

Confuses the graphs of  direct and inverse proportion

Believes a less steep gradient corresponds to a faster speed

Believes a direct proportion graph can have a negative gradient

When reading value from graph, reads from the wrong axes.

Believes a reciprocal graph can be a straight line

Does not realise that gradient represents rate of change

Believes when you plot a linear sequence it doesn't give a straight line

Believes a cubic equation will have a linear graph

Believes that graphs of inverse proportion meet the axes

Thinks depth is marked on the cross section of a prism

Struggles to visualise elevations of 3D shapes

Assumes that a graph always starts at the origin

Does not understand that a line graph shows a relationship between 2 variables

Thinks depth always goes front to back in a diagram of a 3D shape

Confuses rest and movement on a distance time graph

Does not recognise that a linear sequence must increase or decrease by same amount

Believes an upward line has a negative gradient

Believes scales on graphs must match on both axes

Struggles to visualise nets of 3D shapes

Believes you cannot have a negative gradient when plotting a linear sequence

Does not know the meaning of the word regular

Believes direct proportion means a 1:1 ratio

Thinks terms in linear sequence are in direct proportion

Thinks the shape of the graph does not affect the area calculation

Believes a quadratic graph can be a straight line

Assumes all journeys of the same distance take the same amount of time

Assumes regular shape refers only to a square

Confuses the dependent and independent variables on the axes of a line graph

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: Horizontal and Vertical Lines ### ConstructName: Recognise the equations of horizontal lines from a diagram ### Question: What is the equation of this line? ![A graph showing a horizontal line going through 4 on the y axis.]()

Confuses the equations of vertical and horizontal lines

Thinks the more digits a number has the greater it is, regardless of place value

When ordering decimals, believes the decimal point is not relevant

When ordering integers, orders from the digits at the end of the number instead of the digits at the start

Believes negative numbers with a larger absolute value are greater

Mixes up smallest and largest when ordering decimals

Does not know how to compare decimal numbers

Orders integers based on first digits without taking into account place value

Does not know the units column is before the decimal point

Chosen the longest value rather than the greatest

When ordering numbers in standard form just looks at the decimal without multiplying by a power of 10

Student thinks that negative decimals behave the same as positive decimals when positioning a decimal between two consecutive negative integers, placing the decimal with a greater absolute value to the right.

Does not equate the value of a number with the place value column it is in

When placing a negative number on a number line student assumes numbers with a greater absolute value are placed to the right.

Does not understand that having values after the decimal point makes a number larger than a number with the same units but no decimal places

Believes -a is always smaller than a, ignoring the possibility that a is negative

Does not understand place value after the decimal point

Mixes up smallest and greatest when ordering positive integers

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

Does not understand which place value is the one that determines if a number rounds up or down

Believes digits of a number are read/written from right to left

Mixes up greater than and less than symbols

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

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

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

Does not understand the value of zeros as placeholders

Mixes up colder and hotter when comparing temperatures

Counts on in the wrong direction on a number line

Believes a number written in standard form is smaller than another number written in standard form if the number between 1 and 10 is smaller

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

Counts in the wrong direction on a number line

Believes a number written in standard form with a negative power creates a larger number than that with a positive power

Thinks subtracting a negative makes your answer smaller

Does not understand the meaning of "value"

Does not understand place value within a number

Mixes up smallest and largest when ordering fractions

Identifies wrong place value column after decimal point

Believes that fractions with larger denominators are greater in size

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

Believes you subtract the smaller digit from the larger digit in column subtraction, regardless of position

Assumes that a list is given in size order

Ordered fractions by comparing denominators rather than using equivalent fractions.

Heads towards zero when subtracting from a negative

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

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

Changes wrong place value column when rounding

Thinks decimal place value works the same as integer place value, ones will follow tenths

Rounds to the wrong degree of accuracy (rounds too little)

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 Fractions ### ConstructName: Add proper fractions with the same denominator ### Question: Calculate:
\(
\frac{2}{7}+\frac{1}{7}
\)

When asked to add fractions, multiplies instead

Does not recognise when simplifying a fraction may be helpful

Does not understand how to simplify fractions

Does not recognise when a fraction can be simplified

Does not understand a percentage is out of 100

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

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

Only reads the percentage symbol as applying to the units digit

Does not realize we can write a division as a fraction

Believes percentages cannot be converted into fractions

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

Does not know how to read information off a pie chart

Does not know how to simplify algebraic fractions

Does not understand equivalent fractions

Unable to equate a division and multiplication to a single division

Does not understand how to find percentages of an amount

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

Does not know how to convert percentages with a decimal point into decimals

Does not understand changing decimals to fractions

Thinks they can convert a 2 digit percentage to a fraction by writing one number over the other.

Thinks that a unit fraction can be simplified further

Unable to equate two divisions with a single division

Does not understand the relationship between a ratio and fraction

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

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

Does not recognise difference of two squares

Thinks you need to just remove the percentage sign to convert from a percentage to a decimal

Does not understand that the numerator and denominator of fractions represent groupings and have the same order of priority as brackets

Does not realise ratio in worded problem can be simplified

Does not understand the value of zeros as placeholders

Does not realize we can simplify a term that involves powers

Assumes fractions with prime numerators cannot be simplified

Does not know that you can have a decimal percentage

Assumes that converting a decimal percentage to a fraction, they can just convert the decimal to a fraction, instead of writing the decimal over 100.

Believes you cannot find the angle size in a pie chart when given its percentage value

When converting a percentage to a decimal, just writes all the digits after a decimal point.

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

Assumes decimal point can be ignored when given a decimal percentage to convert to a fraction

Does not realise you can use equivalent fractions to break fractions up into smaller divisions

Does not recognise factorising a quadratic

Thinks they can convert percentages to fractions by writing the percentage value as both the numerator and 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: Substitution into Formula ### ConstructName: Substitute positive integer values into expressions involving powers or roots ### Question: When \( h=10 \)
Which of the following pairs of statements is true?

Answers order of operations questions with brackets as if the brackets are not there

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

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

Thinks rule for calculating interior angles only applies to regular polygons

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

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

Thinks the interior angle of a regular polygon is 10n

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

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

Does not know how to calculate the sum of interior angles

Thinks the interior angles of any polygon add up to 360

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

Does not understand what an interior angle is

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

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

Does not recognise that interior and exterior angles are supplementary

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

Does not know the meaning of the word regular

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

Believes interior angles are outside a shape

Does not consider that all three angles in a triangle sum to 180 degrees

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

Does not know that angles in a triangle sum to 180 degrees

Assumes regular shape refers only to a square

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

Only multiplies the first term in the expansion of a bracket

Does not recognise parallel lines angle rules applied inside a shape

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

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

Does know know equilateral means the same as regular

Thinks the value of an exterior angle is 360 degrees

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

Confuses interior angles with exterior angles

Does not know that angles on a straight line sum to 180 degrees

Forgets to subtract sum of angles we know from 180 to find missing angle in a triangle

Identified the interior not exterior angle at a point

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

Only multiplies second term in the expansion of a bracket

Thinks only one variable can introduce bias

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

Finds interior angle rather than given angle

Believes the constant only gives the y-intercept for linear graphs

Thinks only angles on parallel lines can be alternate

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

Does not recognise that addition is the inverse operation of subtraction

Does not understand the concept of inverse operations in algebra

Assumes a fact without considering enough examples

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: Reflection ### ConstructName: Reflect an object or a point in a diagonal line with the line of reflection drawn, where the line of reflection lies on the edge or outside the object ### Question: The triangle is reflected in the dashed line.
What are the new coordinates of point \( \mathrm{P} \) ? ![A coordinate grid with a reflection line drawn and a triangle. The reflection line passes through the coordinates: (-1,-1) (0,0) (1,1) (2,2) etc. The triangle has coordinates: (2,2) (3,3) and (4,2). P is the coordinate (4,2).]()

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

Mixes up greater than and less than symbols

Does not understand the 'less than' inequality notation

Assumes that a list is given in size order

When representing an inequality on a numberline, assumes the filled and unfilled circles have the opposite meaning.

When ordering integers, orders from the digits at the end of the number instead of the digits at the start

Believes the sides of an inequality can be switched without changing the direction of the sign

Believes negative numbers with a larger absolute value are greater

Confuses the shaded and unshaded regions on an inequality diagram

When asked to find data points below a certain value, gives the number of data points above this value

Counts on in the wrong direction on a number line

Confuses inequality symbols with equality

Counts in the wrong direction on a number line

When placing a negative number on a number line student assumes numbers with a greater absolute value are placed to the right.

When asked to find data points above a certain value, gives the number of data points below this value

Believes that fractions with larger denominators are greater in size

When using inequality notation, thinks "greater than/less than or equal to" symbols are instead the "greater than/less than" symbols and vice versa

Believes inequalities cannot be combined

Believes that 'less than' includes the number

Thinks subtracting a negative makes your answer smaller

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

Believes a smaller number of shaded parts represents a smaller fraction

Believes greater than/less than symbols include end values

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

Mixes up the inequality signs in an error interval

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

Believes an upward line has a negative gradient

When ordering decimals, believes the decimal point is not relevant

Thinks a dashed line marking a region means the line is included

Selects y>a solutions instead of y<a solutions for a quadratic inequality

Mixes up smallest and greatest when ordering positive integers

Mixes up colder and hotter when comparing temperatures

Confuses the union and intersection of inequalities

Believes reading from a cumulative frequency diagram will give the frequency above a certain value, rather than below

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

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

Student thinks that negative decimals behave the same as positive decimals when positioning a decimal between two consecutive negative integers, placing the decimal with a greater absolute value to the right.

Believes inequalities can only include integers

Believes -a is always smaller than a, ignoring the possibility that a is negative

Believes digits of a number are read/written from right to left

Cannot evaluate if an estimate is an underestimate or overestimate

Heads towards zero when subtracting from a negative

Believes rounding numbers down would give an overestimate

Thinks a greater number of shaded parts represents a greater 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: Equation of a Circle ### ConstructName: Given a sketch of a circle with centre at the origin, write its equation in the form x² + y² = r² ### Question: What is the equation of this circle? ![A set of axes with a circle drawn. The circle goes through the points (4,0), (0, -4), (-4,0) and (0,4). ]()

Mixes up squaring and multiplying by 2 or doubling

Does not find a common denominator when adding/subtracting fractions

Forgets to change the numerators when finding equivalent fractions

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

Does not understand how to maintain equivalence in fractions

When subtracting fractions, subtracts the numerators and denominators

Does not find a common denominator when comparing fractions

Does not understand equivalent fractions

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

Does not understand how to subtract algebraic fractions

When subtracting fractions, divides the numerators

Removes the common denominator when making equivalent algebraic fractions

Subtracts numerators when dividing fractions with a common denominator

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

Incorrectly cancels what they believe is a factor in algebraic fractions

Subtracts the smallest digit from the largest digit

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

Believes that fractions with the same denominator are equal

Believes equivalent fractions are formed with addition not multiplication

When subtracting fractions, subtracts the numerators and multiplies the denominators

Multiplies the numerators when adding fractions with a common denominator

When adding fractions, adds the numerators and denominators

Thinks the fraction bar means subtract rather than divide

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

Adds instead of subtracts when asked to find the difference

Thinks the subtraction sign means multiply

Carries out operations from left to right regardless of priority order

Uses numerators to convert fractions to ratio despite different denominators

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

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

Thinks the subtraction sign means divide

Confused mixed numbers with improper fractions

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

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

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

Does not know that 7 and -7 are different

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

Ignores the negative signs when adding positive and negative numbers

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: Interpret a pie chart ### Question: A darts player hits his target \( 40 \% \) of the time. Which pie chart represents his hits and misses?

Does not know how to read information off a pie chart

Does not understand the first significant value is the first non-zero digit number

Identifies second decimal place value when asked for the second significant figure

Identifies first decimal place value when asked for first significant figure

Gives the first significant figure when asked for the second

Rounds to wrong degree of accuracy (decimal places rather than significant figures)

Gives the second significant figure when asked for the first

Counts leading zero as a significant figure

Rounds to the wrong degree of accuracy (significant figures rather than decimal places)

Rounds to the wrong degree of accuracy (rounds too much)

Rounds to the wrong degree of accuracy (rounds too little)

Forgotten placeholders when rounding to significant figures

Does not understand which place value is the one that determines if a number rounds up or down

Does not understand the value of zeros as placeholders

Changes wrong place value column when rounding

Rounds down instead of up when next digit is 5

When ordering integers, orders from the digits at the end of the number instead of the digits at the start

Believes digits of a number are read/written from right to left

Rounds to the nearest integer rather than one significant figure

Rounds to the nearest hundredth, rather than one significant figure

Orders integers based on first digits without taking into account place value

Rounded to wrong degree of accuracy (2sf not 1sf)

Does not understand the meaning of "value"

Rounds incorrectly by changing multiple place values

Does not know the units column is before the decimal point

When ordering decimals, believes the decimal point is not relevant

Thinks the more digits a number has the greater it is, regardless of place value

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

Identifies wrong place value column after decimal point

Believes you subtract the smaller digit from the larger digit in column subtraction, regardless of position

Rounds up rather than to one significant figure

When rounding to decimal places and the only digits after the decimal point will be zeros, just gives the integer as the answer.

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

When rounding, believes that if the number, after the required degree of accuracy, is a 5 or below you round down

Mixes up greater than and less than symbols

Believes rounding to 1 significant figure means rounding to the nearest integer, regardless of the value of the number

Believes that rounding must change the number

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

Thinks the first term in a sequence is the term before the sequence begins (0th term)

Counts on in the wrong direction on a number line

Thinks that the lower and upper bound for a truncated number can be found in the same was as when it has been rounded

Mixes up smallest and largest when ordering decimals

Does not equate the value of a number with the place value column it is in

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

Gives a rounded whole number instead of a decimal

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 \( 21 \times 12 \) mentally Tom does \( 12 \times 7 \) and then multiplies his answer by \( 3 \) Katie does \( 21 \times 6 \) and then doubles her answer Who would get the correct answer?

Does not correctly apply the distributive property of multiplication

Thinks the sum of 2 numbers, each squared first, is the same as adding the two numbers and then squaring the total

Square roots when asked for second multiple

Believes a power of 2 has no impact on its value

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

Believes squaring is the same as adding two

Believes the square of a negative will also be negative

Squares when asked to find the square root

Mixes up square numbers and cube numbers

Does not think a square root can be negative

Mixes up square numbers and prime numbers

Only looks at the difference between one pair of terms when discerning the pattern

Halves when asked to find the square root

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

Believes the difference between triangle numbers remains the same

When comparing non like surds, does not square the surds

Adds instead of subtracts when asked to find the difference

Mixes up square numbers and even numbers

Thinks you can find missing values in a given table by treating the row as linear and adding on the difference between the first two values given.

Does not link Pythagoras Theorem to finding distance between two points

Believes square numbers are triangle numbers

When adding roots, adds the numbers under the roots

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

Thinks you can find missing values in a given table by treating the row as a sequence of consecutive numbers

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

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

Mixes up squaring and multiplying by 2 or doubling, and not square rooted

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

When finding the distance between coordinates, includes the starting number

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

Chooses an example when asked for a counter-example

When finding the distance between 2 points, believes the distance is the coordinate of the furthest point from origin

Assumes a fact without considering enough examples

Adds rather than subtracts when using Pythagoras to find a shorter side

Believes the difference between triangle numbers decreases by 1 each time

Believes you can square root a negative number

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

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: Quadratic Equations ### ConstructName: Solve quadratic equations using the quadratic formula where the coefficient of x² is not 1  ### Question: Mo wants to solve this equation using the quadratic formula.
\(
5 h^{2}-3 h-2=0
\)

What total should replace the square? \( \begin{array}{l} \frac{ ?\pm \sqrt{9-(-40)}}{?} \\ = \frac{ ?\pm \sqrt{\square}}{?} \end{array} \)

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

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

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

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

Believes the solution of mx + c = a is x =a

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

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 they solve mx + c = 0 by substituting in x = 0

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

When reading value from graph, reads from the wrong axes.

Finds the y intercept when asked to read the solution to a quadratic equation from a graph

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

Only gives 1 value of x for a given value of y, when reading from a parabolic graph

Solves for y=0 instead of given y-value when reading from a quadratic graph

Believes that in y=mx+c, m is the x-intercept

When solving simultaneous equations graphically, only gives one of two solutions

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

Stops iteration too early when solving equations

When finding an input of a function machine thinks you apply the operations given rather than the inverse operation.

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

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

Believes the y-intercept of a line is the solution to its equation

When solving simultaneous equations, assumes every variable has equal value.

When asked to read a value from a graph, thinks they give the corresponding value on the other axis, rather than reading from the graph

Heads towards zero when subtracting from a negative

Believes that in y=mx+c, m is the y-intercept

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

Does not reverse the order of operations when solving an equation

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

Believes the equation of a straight line is y=-mx+c

Believes solving an inequality gives a single value as a solution

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

When solving an equation, subtracts instead of dividing

Thinks that in the equation x=a, a is the gradient

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

When solving simultaneous equations graphically, they think the solutions are the x or y intercepts

Believes the solutions of a quadratic equation are the constants in the factorised form

When solving simultaneous equations graphically, they think the solutions are the y-intercepts

Gives an answer of 0 when there are no solutions to an equation.

Includes the x variable when giving the equation of a horizontal line

Thinks x = ? is a horizontal line through ?

Believes subtracting a positive number from a negative number makes the answer less negative

Believes that a number divided by itself equals the original number

Reflects in an axis, instead of the given mirror line

Believes the equation of a straight line is y=-mx-c

Thinks the sign of solutions in a quadratic equation is always the same as the constant term

When solving simultaneous equations graphically, they think the solutions are the x-intercepts

Has used the wrong data point on the graph

Believes substituting the starting value into an iteration formula immediately gives the solution

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