Question: ### SubjectName: Mental Addition and Subtraction ### ConstructName: Complete missing number mental subtraction problems up to 100 ### Question: What should replace the star? \( \bigstar-37=38 \)

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

Counts the starting number when counting on

Believes the number of jumps between two numbers on a numberline represents the sum of those 2 numbers

Counts the starting number when counting backwards

Forgets to add on the starting value when identifying a position on a number line

Believes a gap of 2 is a gap of 1 on a numberline

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

When reading integers on a number line, assumes each dash is 2

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

Assumes that an arrow on a number line starts at zero

Counts in the wrong direction on a number line

Counts on in the wrong direction on a number line

When finding the difference between two marks on a scale, counts lines instead of gaps

When finding the distance between coordinates, includes the starting number

Believes we move down a number line when we add

Does not include zero when counting over zero

Ignores the negative signs when adding positive and negative numbers

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

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

Heads towards zero when subtracting from a negative

When reading integers on a number line, assumes each dash is 10

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

When describing a translation, miscounts when crossing an axis

Forgets to add starting value when calculating new coordinates

when reading a dash on a number line counts 1 dash too few when working out the divisions

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

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

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

Subtracts the smallest digit from the largest digit

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

Reads the position of a number from the space between the line marked, rather than the line marked

Thinks positive or negative sign affects the term before it

Believes the arrow on a number line points to the middle value

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

Ignores position/value of midpoint when trying to label a point on scale

Adds instead of subtracts when asked to find the difference

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

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: Use perimeter to write algebraic expressions ### Question: What is an expression for the perimeter of this rectangle? ![Yellow rectangle with base length a and side length b]()

Fails to use the properties of simple shapes to work out missing lengths

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

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

Has used the wrong data point on the graph

Assumes that a graph always starts at the origin

When finding the difference between two points on a time series graph, only uses one of the points.

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

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

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

When reading graph, chooses nearest marked value on axis instead of reading accurately

Believes a direct proportion graph does not have to start at the origin

Believes scales on graphs must match on both axes

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

Does not check the labelling of the axes

Believes direct proportion means a 1:1 ratio

Does not understand the command word 'difference'

Believes change in y for gradient is always calculated from the origin, regardless of the y-intercept of the line

Believes the line y = mx - c will cross the y axis at c

Does not recognise the origin as the starting point for coordinates

Assumes every dash on a scale represents an increase of 1

Assumes the scale is the same when comparing two similar diagrams

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

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

Believes it's not possible to solve simultaneous equations graphically

Counts squares instead of the correct scale when plotting coordinates

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

Believes a direct proportion graph is a horizontal line

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

Believes the gradient is where the line intersects the y axis

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

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

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

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

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

Believes that graphs of inverse proportion meet the axes

When comparing data, only compares one category of data rather than comparing the totals

Does not understand the difference between a value on a graph and the gradient of a graph

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

When finding the distance between coordinates, includes the starting number

Thinks that the scale on axes has to match with the number of squares

When describing a translation, miscounts when crossing an axis

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

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

Not considered the y value when reading coordinates

Believes that the y intercept is where y = 0

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

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


Question:
### SubjectName: Expanding Double Brackets ### ConstructName: Expand two brackets with linear terms in the form (ax + b)(cx + d) ### Question: Alison is expanding these two brackets.
What should she get when she multiplies the two terms indicated by the arrows? ![The two brackets are (5p-4)(2p-3). The arrows are pointing at the -4 in the first bracket and the -3 in the second bracket.]()

Believes multiplying two negatives gives a negative answer

Does not understand bar modelling in algebra

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

Does not understand what each part of a bar model represents as a fraction

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

Believes the number of bars gives the total on a bar chart

Confuses section and whole when using a bar model

Does not know how to use a bar model to write division facts

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

Does not take into account the whole amount when using a bar model to find a fraction

Only multiplies the first term in the expansion of a bracket

Does not realise that two letters together are multiplying

When reading a composite bar chart, just reads total height of bar

Only multiplies second term in the expansion of a bracket

Thinks the longest bar has the highest frequency on a pictogram, even if the symbols aren't evenly spaced

Does not understand use of the counters to represent algebra

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

Believes class width represents area of the bar in a histogram

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

Does not know what the symbol for equal lengths means

Believes number of squares in a bar gives frequency on a histogram

Writes the variable before the coefficient in an algebraic term

Assumes the scale is the same when comparing two similar diagrams

Does not understand parts must be equal in a fraction diagram

Assumes value of symbols in a pictogram without checking the key

Only multiplies the numerical terms when expanding a bracket

Does not know the meaning of the identity symbol

Believes class width multipled by frequency gives the area of the bar in a histogram

Does not understand when like terms can be collected

Ignores coefficients when forming an expression from a question

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

Does not know the correct notation for labelling a side length

When interpreting a pictogram, thinks each symbol stands for 1

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

Does not know that a single letter labels a vertex

Does not include brackets when attempting to multiply an expression

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

Counts partial symbols as if they were whole on pictograms

Does not understand how to multiply algebraic terms

Confuses collecting like terms with multiplication

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

Mixes up total and part when forming expressions from worded problems

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

Thinks the smallest bar in a bar chart is the mode

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


Question:
### SubjectName: Length Scale Factors in Similar Shapes ### ConstructName: Solve length problems involving similar shapes where the missing length is the shorter one ### Question: These two shapes are similar (not drawn to scale).
What should replace the star? ![Two scalene triangles. The bigger one has sides 9 cm and 7.5 cm. On the smaller triangle, the side corresponding to the 9 cm is 6 cm, and the side corresponding to the 7.5 cm is marked with a star.]()

Does not understand how to find a scale factor

Does not understand how to apply sohcahtoa

Believes the adjacent side is where the opposite side is in a right angled triangle

Confuses order of sides in a trigonometric ratio

Tries to apply sohcahtoa to a non right angled triangle

Believes the opposite side is where the adjacent side is in a right angled triangle

Misremembers the exact values of trigonometric functions at standard angles

Forgets to apply inverse trigonometric function when finding an angle

Believes 0.5absin(c) requires only three sides (potentially confused with cosine rule)

Believes the sin graph relates to the cos function

Is not able to identify a situation where Pythagoras' Theorem is needed

Believes any of the angles can be used in the trigonometric formula for area of a triangle

Has an incorrect assumption about the definition of a reciprocal

Does not know that a right angle is 90 degrees

Does not link Pythagoras Theorem to finding distance between two points

Subtracts rather than adds when using Pythagoras to find the hypotenuse

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

Does not know the meaning of square sign for right angle

Forgets to include a marked right angle when performing angle calculations

Does not realise that the order is important in a ratio

Believes the sin graph relates to the tan function

Gives 3-letter angle notation in the wrong order

Does not know the correct notation for labelling a side length

Does not rearrange the Pythagoras' Theorem formula accurately

Only uses the closest angle to the unknown when using the angles in a straight line relationship

Believes unrelated acute/obtuse angles in a diagram are supplementary

Does not know the meaning of corresponding

Does not understand that base angles in an isosceles are equal

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

Believes all triangles have a right angle

Does not know the definition of a rational number

Does not know the word and/or does not know the function of a compass

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 different denominators which do not share a common factor ### Question: \(
\frac{1}{3}+\frac{1}{10}=\frac{\bigstar}{30}
\)
What should replace the star?

Does not find a common denominator when adding/subtracting fractions

Mixes up square numbers and even numbers

Believes even numbers can have odd multiples

Believes two odd numbers multiply to give an even number

Believes all multiples of odd numbers are odd

Does not recognise that the intersection of a Venn diagram represents those belonging to both sets

Chooses an example when asked for a counter-example

Gives the intersect value of a Venn diagram when asked to find the value of a set only

Believes the intersection on a Venn diagram represents the total of one set

Includes the intersection of a Venn diagram when asked to find the value of a set only

Does not understand that 2n represents an even number

Believes that prime numbers cannot be even

Believes the number outside the circles in a Venn represents the intersection

Assumes a fact without considering enough examples

Thinks numbers are a multiple of their index

Believes the square of a negative will also be negative

Mixes up square numbers and prime numbers

Mixes up squaring and multiplying by 2 or doubling

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

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

Only considers elements in one set when asked to find the intersection of multiple sets

Believes adding b to any integer gives a multiple of b

Believes square numbers are triangle numbers

Does not understand how to answer generalised statements (eg. by testing values to check for always, sometimes, never)

Does not think a number can be a multiple of itself

Believes the intersection in a prime factor venn diagram does not contribute to the size of the number represented by a circle

Thinks the term in a sequence is double the position value

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

Believes that the complement of a set includes any elements which are also in another set.

Does not know the definition of a prime number

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

Believes the total of the circle in a Venn should be included when intersections should be omitted

Thinks you cannot find the range if there are an even number of values

When finding the median for an odd numbered data set, find the midpoint of two values rather than giving the middle value

Forgets to include the intersection value when asked for total value of a set in a Venn diagram

believes the complement of a set consists of elements which only belong to one set

Does not know the definition of a composite number

When finding the median from a even dataset does not understand we must find the midpoint of the two values in the middle

Does not know the meaning of consecutive

Believes that the complement of a set in a triple Venn consists of the elements that only belong to one other set

Does not know the meaning of the word identity

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: Rotation ### ConstructName: Rotate an object or point 900 clockwise or anticlockwise about a point, where the point lies on the edge or outside of the object ### Question: John rotates rectangle \( A B C D 90^{\circ} \) clockwise from the centre \( (3,2) \). What error has he made? ![A positive, coordinate grid with two rectangles drawn on it. The rectangle labelled ABCD has the coordinates: A(1,3) B(1,5) C(2,5) and D(2,3). The other rectangle has the coordinates: (4,3) (6,3) (6,2) and (4,2).]()

Does not recognise the notation for parallel sides

Does not know the meaning of the word parallel

Thinks that if two lines do not intersect on a diagram then they will never intersect and are therefore parallel

Does not know what dashes on the sides of shapes mean

Confuses the sign for parallel lines as being dashes

Does not know parallel lines have equal gradients

Does not know that co-interior angles must be in parallel lines

Believes the arrows for parallel lines mean equal length

Does not recognise parallel lines angle rules applied inside a shape

Does not know the meaning of corresponding

Believes two lines of the same length are parallel

Does not know what the symbol for equal lengths means

Does not understand the diagram given in the question

Describes lines when asked to describe angles

Does not know the properties of a rectangle

Does not know how to compare size of angles on a grid without a protractor

Does not know the meaning of perpendicular

Thinks sign for parallel lines just means opposite

Does not recall rectangle side length property

Does not know the properties of a parallelogram

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

Confuses the meaning of parallel and perpendicular

Cannot identify a line of symmetry in a simple shape

Believes lines are parallel if they have the same y intercept

Does not realise similar has a specific mathematical meaning

Believes parallel is the term used to describe two lines at right angles

Confuses the signs for straight lines as being dashes

Believes it's not possible to solve simultaneous equations graphically

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

Believes parallelogram is the term used to describe two lines at right angles

Does not understand how to find a scale factor

Does not realise the lines at an intersection have to be straight in order to show vertically opposite angles

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

Believes mirror line must lie on a grid line, not in between grid lines

Fails to reflect across reflection line without a diagram

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

Believes any lines that cross are perpendicular

Believes parallel lines have gradients of opposite signs

Believes only angles on parallel lines can be called "corresponding"

Does not know the correct notation for labelling a side length

Does not recognise the properties of a kite

Does not recognise that a shape translated would not change orientation

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: Multiplying and Dividing with Decimals ### ConstructName: Divide two decimals with a different number of decimal places ### Question: \( 70.04 \div 0.1= \)

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

Does not understand that area under a speed time graph gives distance because distance = speed x time

Believes the length of the line in a velocity time graph gives the distance (rather than the area underneath that line).

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

Believes distance is the area under the graph on a displacement time graph

Believes that when finding area under graph you can use the upper y value rather than average of upper and lower

When finding the difference between two points on a time series graph, only uses one of the points.

Adds when calculating area instead of multiplying

Believes that addition should be used when finding the difference between two values on a time series graph

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

Believes scales on graphs must match on both axes

Believes speed is distance multiplied by time

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

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

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

Believes the gradient of a speed-time graph is the distance

Calculates perimeter when asked for area

Believes the gradient of a speed-time graph is the speed

When calculating perimeter, only adds the marked lengths

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

Confuses displacement-time graphs with velocity-time graphs

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

Does not divide by 2 when finding midpoint

Adds when calculating volume instead of multiplying

When reading graph, chooses nearest marked value on axis instead of reading accurately

Added rather than multiplied by a scale factor

Finds the difference in a coordinate pair rather than the midpoint

Has used the wrong data point on the graph

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

Believes a downward slope on a speed-time graph shows positive acceleration

Does not understand the difference between a value on a graph and the gradient of a graph

Forgets to add starting value when calculating new coordinates

Believes midpoint calculation is just half of the difference

Believes midpoints are plotted on histograms

Assumes every dash on a scale represents an increase of 1

Does not have when calculating the area of a triangle

Assumes grouped data is evenly distributed within each group

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

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

Believes the midpoint of grouped data is equivalent to the group width

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

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

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


Question: ### SubjectName: Percentages of an Amount ### ConstructName: Express one quantity as a percentage of another mentally ### Question: What is \( 50 \mathrm{~m} \) as a percentage of \( 5 \mathrm{~km} \) ?

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

Does not recall rectangle side length property

Does not know the properties of a rectangle

Does not know the diagonals of a square bisect at right angles

Does not know the properties of a parallelogram

Does not know the properties of a rhombus

Does not believe the length of a diagonal line can be worked out based on horizontal and vertical measurements

Does not know that a right angle is 90 degrees

Believes a parallelogram cannot have right angles

Does not know what dashes on the sides of shapes mean

Does not recognise the properties of a kite

Does not know the meaning of perpendicular

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

Does not understand the diagram given in the question

Believes isosceles triangles cannot have right angles

Does not know a rectangle is a type of parallelogram

Does not know the meaning of square sign for right angle

Thinks that perpendicular lines do not meet at 90 degrees

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

Does not recognise parallel lines angle rules applied inside a shape

Does not know the properties of an isosceles triangle

Does not understand how to answer generalised statements (eg. by testing values to check for always, sometimes, never)

Does not know the meaning of the word regular

Does not know a rhombus is a type of parallelogram

Believes any lines that cross are perpendicular

Believes particular is the term used to describe two lines at right angles

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

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

Assumes a fact without considering enough examples

Fails to reflect across reflection line without a diagram

Does not realise the lines at an intersection have to be straight in order to show vertically opposite angles

Believes a rectangle has one pair of equal sides

Is not able to identify a situation where Pythagoras' Theorem is needed

Believes a rectangle has four equal sides

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

Thinks rule for calculating interior angles only applies to regular polygons

Thinks that if two lines do not intersect on a diagram then they will never intersect and are therefore parallel

Does not know that co-interior angles must be in parallel lines

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

Given the length of the sides, still does not recognise there is enough information to find the area

Believes that angles that look like they are a right-angle are always 90 degrees

Fails to use the properties of simple shapes to work out missing lengths

Does not know the definition of the range

Does not understand the meaning of the word commutative

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


Question: ### SubjectName: Written Division ### ConstructName: Interpret and carry out mental division using a bar model ### Question: Find the value of \( P \). ![Horizontal green bar divided into 5 equal sections. Each section is labelled with a "P". Curly bracket shows that 3 of the sections sum to 615.]()

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

Has multiplied the base number and the power

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

Believes a power of 2 has no impact on its value

Writes a variable to the power of n as nx

Believes cubing a number means you multiply it by 3

Believes squaring a fraction requires you to double both parts

Believes the square of a negative will also be negative

Has added the powers rather than multiplying them

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

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

Multiplied the powers rather than added them

Assumes they multiply by the numerator of fractional index

When substituting a value into a non-linear term with a coefficient, multiplies by the coefficient then calculates the power.

Has completed only one of the two operations.

Believes term to term rule is applied to position number, not previous term

Believes multiplying two negatives gives a negative answer

Subtracted base numbers as well as powers

Thinks index notation represents repeated addition rather than repeated multiplication

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

Believes squaring is the same as adding two

Believes division is carried out before a power

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

Believes the inverse of square rooting is doubling

Believes squaring a negative number just changes the sign

Thinks numbers are a multiple of their index

Repeats the digits twice when squaring a number

Carries out operations from left to right regardless of priority order

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

Adds instead of multiplying when expanding bracket

Thinks halving is the same as multiplying by 2

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

Forgot to square both parts of the 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: Properties of Quadrilaterals ### ConstructName: Recall and use the intersecting diagonals properties of a rectangle ### Question: The diagonals of a rectangle with different length sides \(_____________\) intersect at right-angles ![A rectangle]()

Does not know the properties of a rectangle

Believes speed is distance multiplied by time

Thinks the units for speed should not contain a measure of time

Assumes the denominator is divided by the numerator

Believes that a fraction means dividing the denominator by the numerator

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

Believes gradient = change in x/change in y

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

When adding fractions, adds the numerators and multiplies the denominators

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

Believes gradient is the difference between the change in x and the change in y

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

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

When multiplying fractions, multiplies the denominator

When adding fractions, adds the numerators and denominators

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

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

When subtracting fractions, subtracts the numerators and multiplies the denominators

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

When dividing fractions, does the reciprocal of both fractions

Believes class width = frequency density

When finding the gradient from the equation of a line in the form ax + by = c, believes b/a is the gradient

Mixes up the value of two terms when substituting

Thinks the fraction button finds the reciprocal

Believes you divide 1 by the number of values, rather than the total, when finding the mean

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

Uses term-to-term rule as numerical part of nth term rule

Gets their constants and coefficients the wrong way around when writing a formula from a given situation

Uses dividing fractions method for multiplying fractions

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

Believes gradient at a point = y coordinate/x coordinate

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

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

Multiplies instead of divides when division is written as a fraction

Divides when converting to a smaller unit

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

Believes they always divide by the conversion factor when converting units

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

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

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


Question: ### SubjectName: 2D Names and Properties of Shapes-Others ### ConstructName: Understand the terms equilateral and regular, and recognise the notation for regular polygons ### Question: Tom and Katie are discussing regular polygons. Tom says this is a regular polygon ![A triangle with 3 equal sides and 3 equal angles]() Katie says this is a regular polygon ![A parallelogram with 2 pairs of parallel sides marked]() Who is correct?

Does not know the meaning of the word regular

When finding the gradient from the equation of a line in the form ax + by = c, believes b/a is the gradient

Believes that for a line written in the form ax + by = c, a is the gradient

When solving an equation where there is a negative sign in front of the variable at the final stage, believes they can ignore it and treat it as if it's positive

Believes dividing a negative by a positive gives a positive answer

When given the equation of a straight line, believes the first term always gives the gradient, even when the first term is actually c

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

Believes dividing a positive by a negative gives a positive answer

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

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

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

Believes that the coefficient of x represents the gradient even when a line is not in the form y = mx+c

Believes that when eliminating a variable, if the signs of the terms with matching coefficients are the same, we add the equations.

Ignores the negative signs when adding positive and negative numbers

Believes that when eliminating a variable, if the signs of the terms with matching coefficients are the same, we do the operation inverse to the sign (- if positive, + if negative)

Believes that when eliminating a variable, if the signs of the terms with matching coefficients are the same, we do the operation matching the sign (+ if positive, - if negative)

Believes multiplying a positive by a negative gives a positive answer

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

When solving an equation, believes that when they get to ax = b, b is the solution

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

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

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

Believes a downward line has a positive gradient

When solving an equation, adds instead of dividing

Believes the constant gives the y-intercept even when a line is not written in the form y = mx + c

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

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

Believes gradient = change in x/change in y

When solving simultaneous equations, finds the correct numbers to transform the equations, but uses them the wrong way around.

When collecting like terms, assigns positivity or negativity incorrectly.

Believes switching the x and y variables in a linear equation gives a perpendicular line

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

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

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

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

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

Does not reverse the order of operations when solving an equation

Believes adding two negatives gives a positive answer

Believes the gradients of perpendicular lines are reciprocals of the same sign

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

Believes that in y = mx+c, c is the gradient

Ignores negative sign when expanding bracket

When solving an equation, subtracts instead of dividing

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

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

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

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

Confuses the inverse operations of addition and subtraction when rearranging equations

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


Question: ### SubjectName: Converting between Decimals and Percentages ### ConstructName: Convert decimals less than 1 with 3 or more decimal place to percentages ### Question: How do you write \( 0.909 \) as a percentage?

Does not understand the value of zeros as placeholders

Does not know the definition of term or term-to-term rule

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

Describes position to term rule using term to term rule

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

Goes the wrong direction in the sequence when identifying term-to-term rule

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

Describes term to term rules using position to term rules

Thinks n=0 for the first term in a sequence

Thinks the first term of a sequence must be 1

Uses term-to-term rule as numerical part of nth term rule

When finding the nth term of a linear sequence, thinks the the first term is the coefficient in front of n.

When asked for a specific term in a sequence just gives the next term

Identifies the first term rather than the next term in a sequence

Uses first term as numerical part of nth term rule

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

Does not know how to calculate the position of a term in a sequence

Believes nth term rule can be given in words

Uses only the first two terms of a sequence to work out a term-to-term rule

Mixes up term in a sequence and position of a term

Thinks you subtract rather than add when finding the previous term in a descending linear sequence

Believes term to term rule is applied to position number, not previous term

When asked for a specific term in a sequence, just gives the position number

Identifies how much repeated use of the term-to-term rule changes a number but does not apply to the sequence

Identifies the term-to-term rule rather than the next term in a sequence

Believes they can find the nth term based on the first term only

When asked for a specific term in a sequence gives a different term, which is not the next

Believes the nth term of a descending sequence will have a positive coefficient of n

Thinks the nth term is when the value of the term is n

Thinks terms in linear sequence are in direct proportion

Believes the position number in a sequence does not have to be an integer

Thinks the term in a sequence is double the position value

When given a term to term sequence with multiple steps, applies one step per term

Only uses two terms of a sequence to work out a term-to-term rule

Uses difference as index instead of coefficient in nth term rule

Believes that in the nth term of a descending sequence, b is the difference between the term and the positive multiple of n

Believes the subscript in iteration notation is the value to be substituted

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

Does not include zero when counting over zero

Believes you multiply the constant difference by the term number to get the term value

Does not know how to find the next term in a sequence

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

When finding the differences between terms in a sequence, believes they can do so from right to left

When using a function machine, confuses the input and the output.

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

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

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). ]()

Confuses the radius with the radius squared in the equation of a circle

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: Quadratic Sequences ### ConstructName: Given a description of an ascending sequence using quadratics in words, write some terms of that sequence ### Question: ![Square and then subtract \( 4 \)]() The first three terms of which sequence below is produced by this term-to-term rule?

Describes term to term rules using position to term rules

Only multiplies the first term in the expansion of a bracket

Only multiplies second term in the expansion of a bracket

Only multiplies the numerical terms when expanding a bracket

Ignores negative sign when expanding bracket

Does not understand equivalent fractions

Adds instead of multiplying when expanding bracket

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

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

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

Does not include brackets when attempting to multiply an expression

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

Has removed brackets but not performed the operation

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

When subtracting fractions, subtracts the numerators and denominators

Does not understand how to subtract algebraic fractions

When subtracting fractions, subtracts the numerators and multiplies the denominators

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

Incorrectly cancels what they believe is a factor in algebraic fractions

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

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

Thinks you can ignore the variable when simplifying algebraic fractions

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

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

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

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

Does not understand how to maintain equivalence in fractions

Does not apply the inverse function to every term in an expression

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

Has completed only one of the two operations.

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

Thinks only one variable can introduce bias

Assumes any variable in a sum on the numerator can be divided by the denominator

Thinks the subtraction sign means multiply

Thinks terms in linear sequence are in direct proportion

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

Believes they must expand brackets before solving an equation

Believes multiplying two negatives gives a negative answer

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

Removes the common denominator when making equivalent algebraic fractions

Does not know how to simplify algebraic fractions

Forgets to change the numerators when finding equivalent fractions

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

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

Multiplies all terms together when simplifying an expression

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

When factorising, finds a factor that goes into only the first term of the 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: BIDMAS ### ConstructName: Insert brackets to make a calculation correct ### Question: \[
12 \times 6-1+2
\]
Where do the brackets need to go to make the answer equal \( 36 \) ?

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

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

When asked for a specific term in a sequence just gives the next term

When asked for the next term in the sequence adds one to the last term given, despite the sequence not going up in ones.

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

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

When asked for a specific term in a sequence gives a different term, which is not the next

When asked for the next term in the sequence subtracts one from the last term given, despite the sequence not going down in ones.

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

Thinks you subtract rather than add when finding the previous term in a descending linear sequence

Does not understand that adding on to a multiple can give you the next multiple

Thinks the term in a sequence is double the position value

Does not know how to find the next term in a sequence

Believes 'inclusive' does not include the end value

Does not know the meaning of consecutive

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

Believes that 'less than' includes the number

Thinks the remainder is the difference between the dividend and the next biggest multiple

Believes there is a multiplicative relationship between values of terms in a linear sequence e.g. 10th term is double 5th term

Thinks terms in linear sequence are in direct proportion

Identifies how much repeated use of the term-to-term rule changes a number but does not apply to the sequence

Believes the position number in a sequence does not have to be an integer

When asked for a specific term in a sequence, just gives the position number

Identifies the first term rather than the next term in a sequence

Believes that in the nth term of a descending sequence, b is the difference between the term and the positive multiple of n

Identifies the term-to-term rule rather than the next term in a sequence

Thinks positive or negative sign affects the term before it

Believes the position of a sequence cannot be an integer

Believes greater than/less than symbols include end values

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

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

Does not know the definition of term or term-to-term rule

Believes adding b to any integer gives a multiple of b

Counts the starting number when counting backwards

Does not notice all changes in a sequence of visual patterns

Believes inequalities can only include integers

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

Thinks it can't be called a sequence with a repeated number at the start

Counts on in the wrong direction on a number line

Counts in the wrong direction on a number line

Believes the nth term of a descending sequence will have a positive coefficient of n

Thinks the nth term is when the value of the term is n

Thinks the positive and negative terms in a sequence will have the same absolute values

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

Assumes that a list is given in size order

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


Question:
### SubjectName: Adding and Subtracting Fractions ### ConstructName: Add proper fractions with the same denominator ### Question: Work out:
\(
\frac{3}{10}+\frac{5}{10}
\)
Write your answer in its simplest form.

When adding fractions, adds the numerators and denominators

Does not find a common denominator when adding/subtracting fractions

Forgets to change the numerators when finding equivalent fractions

Does not understand how to maintain equivalence in fractions

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

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

Does not find a common denominator when comparing fractions

Does not understand equivalent fractions

When subtracting fractions, subtracts the numerators and denominators

Removes the common denominator when making equivalent algebraic fractions

Does not understand how to subtract algebraic fractions

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 numerators when dividing fractions with a common denominator

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

When subtracting fractions, divides the numerators

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

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

Believes equivalent fractions are formed with addition not multiplication

Thinks positive or negative sign affects the term before it

Carries out operations from left to right regardless of priority order

Subtracts the smallest digit from the largest digit

Believes that fractions with the same denominator are equal

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

Multiplies the numerators when adding fractions with a common denominator

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

When “borrowing” during a subtraction problem, does not subtract one from the preceding digit

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

Done a different calculation to the one given

Thinks the fraction bar means subtract rather than divide

Believes that rounding parts of a fraction in different directions means the answer will be the same.

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

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

Denominator used for other parts rather than total parts

Uses numerators to convert fractions to ratio despite different denominators

When subtracting fractions, subtracts the numerators and multiplies the denominators

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

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

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

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: Sequences-Others ### ConstructName: Write the next term of a descending integer linear sequence ### Question: What is the next term in this sequence?

\[
23, \quad 13, \quad 3, \quad \ldots
\]

Identifies the term-to-term rule rather than the next term

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

Cannot identify a fraction on a number line

Believes the arrow on a number line points to the middle value

Ignores position/value of midpoint when trying to label a point on scale

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

Does not understand a denominator represents equal parts

When reading decimals on a number line, assumes each dash is 0.5

Reads the position of a number from the space between the line marked, rather than the line marked

Believes a gap of 2 is a gap of 1 on a numberline

Does not understand equivalent fractions

Estimates a measurement on a diagram rather than working it out

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

Does not understand parts must be equal in a fraction diagram

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

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

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

Does not know how to interpret a fraction worded problem

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

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

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

When finding the median for an odd numbered data set, find the midpoint of two values rather than giving the middle value

Misunderstands dashed lines as being the midpoint

Does not divide by 2 when finding midpoint

Believes midpoints are plotted on histograms

When using interpolation to estimate frequency, assumes half the frequency of the class containing the value will be above/below it, even if it is not the midpoint

Identifies the wrong part of a fraction model as the whole

Believes the frequency of the midpoint of a group is half the total frequency of the group

Believes midpoint calculation is just half of the difference

Forgets to change the numerators when finding equivalent fractions

Believes that fractions with the same denominator are equal

Believes that a fraction means dividing the denominator by the numerator

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

When finding which term represents the median, thinks you divide the total number of terms by 2

Does not understand changing decimals to fractions

Thinks you cannot multiply by a decimal to get equivalent fractions

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

When reading integers on a number line, assumes each dash is 1

Does not find a common denominator when comparing fractions

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

Does not understand what each part of a bar model represents as a fraction

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

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

When finding the median from a even dataset does not understand we must find the midpoint of the two values in the middle

When counting partial symbols on pictogram, uses the fraction rather than the amount it represents

When reading decimals on a number line, assumes each dash is 0.2

Chooses the correct scale when measuring angles on a protractor but starts from the wrong number when the angle is in between two labelled dashes

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 a rhombus from a description of the properties ### Question: Tom and Katie have been asked to give the name of a quadrilateral with four equal sides
Tom says arrowhead
Katie says rhombus
Who do you agree with?

Believes an arrowhead has four equal sides

When asked for the next term in the sequence subtracts one from the last term given, despite the sequence not going down in ones.

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

When asked for a specific term in a sequence just gives the next term

When asked for the next term in the sequence adds one to the last term given, despite the sequence not going up in ones.

Thinks you subtract rather than add when finding the previous term in a descending linear sequence

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

Does not know how to find the next term in a sequence

When asked for a specific term in a sequence gives a different term, which is not the next

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

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

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

Identifies the term-to-term rule rather than the next term in a sequence

Identifies the first term rather than the next term in a sequence

Uses only the first two terms of a sequence to work out a term-to-term rule

Does not know the definition of term or term-to-term rule

Identifies how much repeated use of the term-to-term rule changes a number but does not apply to the sequence

Goes the wrong direction in the sequence when identifying term-to-term rule

Only uses two terms of a sequence to work out a term-to-term rule

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

Counts the starting number when counting backwards

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

Does not notice all changes in a sequence of visual patterns

Believes the nth term of a descending sequence will have a positive coefficient of n

Does not know the meaning of consecutive

Believes that in the nth term of a descending sequence, b is the difference between the term and the positive multiple of n

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

Heads towards zero when subtracting from a negative

When given a term to term sequence with multiple steps, applies one step per term

Describes position to term rule using term to term rule

Does not understand the effect of consecutive subtractions

Thinks the term in a sequence is double the position value

Thinks it can't be called a sequence with a repeated number at the start

Believes a gap of 2 is a gap of 1 on a numberline

Thinks subtracting a negative makes your answer smaller

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

Believes the difference between triangle numbers decreases by 1 each time

Does not understand that adding on to a multiple can give you the next multiple

Believes term to term rule is applied to position number, not previous term

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

Thinks the first term of a sequence must be 1

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

Thinks terms in linear sequence are in direct proportion

Thinks the remainder is the difference between the dividend and the next biggest multiple

Believes an arithmetic sequence can have a common second difference rather than first difference

Counts on in the wrong direction on a number line

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 fractions involving mixed number fractions with different denominators ### Question: \(
2 \frac{1}{3}+\frac{3}{4}=3 \frac{\bigstar}{12}
\)
What should replace the star?

Finds an equivalent improper fraction when asked for a mixed number

Does not recognise factorising a quadratic

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

Incorrectly cancels what they believe is a factor in algebraic fractions

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

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

Cannot identify a common factor when simplifying algebraic fractions

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

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

Thinks you can ignore the variable when simplifying algebraic fractions

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

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

Does not recognise difference of two squares

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

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

When simplifying an algebraic fraction, only looks for number factors

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

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

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

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

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

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

Believes the difference of 2 squares method of factorising also works for the addition of 2 squares

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

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

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

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

Does not know that to factorise a quadratic expression, to find two numbers that add to give the coefficient of the x term, and multiply to give the non variable term

When factorising a quadratic with a coefficient of x squared, believes the signs and/or positions of the constants are interchangeable

Forgets to swap the sign of roots when placing into brackets

When factorising, believes they can choose different factors for each term

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

Thinks a quadratic without a non variable term, can not be factorised

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

Assumes any variable in a sum on the numerator can be divided by the denominator

Does not know how to simplify algebraic fractions

Thinks x^2 - a^2x is a valid form of difference of two squares

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

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

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

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

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

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 the solution of a quadratic is its turning point

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

Misunderstands the concept of common factors in algebra

Does not know how to identify common factors from algebraic terms

Believes numbers are divisible by numbers which sum to their factors

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


Question:
### SubjectName: Expanding Double Brackets ### ConstructName: Expand two brackets with linear terms in the form (x + a)(x + b) ### Question: What should replace the circle when these two brackets are
expanded and simplified?

\(
(p-3)(p+3) \equiv p^{2} \color{purple}\triangle\) \(p\)\(\color{green}\bigcirc
\)

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

Believes the solution of a quadratic inequality is always the interval between the roots

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

Believes inequalities cannot be combined

Confuses the union and intersection of inequalities

Believes any two inequalities can be combined into a double inequality

Mixes up greater than and less than symbols

Thinks two-tailed inequalities can have a constant in the middle

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

Confuses inequality symbols with equality

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

Does not understand the 'less than' inequality notation

Believes solving an inequality gives a single value as a solution

Confuses the shaded and unshaded regions on an inequality diagram

Assumes that a list is given in size order

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

Believes inequalities can only include integers

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

Believes the solution of a quadratic is its turning point

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

Confuses the graph of a linear equation with a quadratic equation

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

Believes greater than/less than symbols include end values

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

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

Believes you can square root a negative number

Believes all quadratic equations have a solution of 0

Believes that the solutions to x<b or x>b also satisfy ax<b or ax>b

Mixes up the inequality signs in an error interval

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

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

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

Believes the solutions to an equation have opposite signs from its x intercepts

Forgets to swap the sign of roots when placing into brackets

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

Does not think a square root can be negative

Believes negative numbers with a larger absolute value are greater

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

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

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

Believes a cubic equation will have a quadratic 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: Construct Triangle ### ConstructName: Construct a triangle using Angle-Side-Angle ### Question: Which of the following best describes how to construct a triangle with side length \( 7 \mathrm{~cm} \) with angles of \( 40^{\degree} \) and \( 50^{\degree} \) either side?

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

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

Does not understand equivalent fractions

Does not understand parts must be equal in a fraction diagram

Does not find a common denominator when comparing fractions

Believes equivalent fractions are formed with addition not multiplication

Believes a smaller number of shaded parts represents a smaller fraction

Forgets to change the numerators when finding equivalent fractions

Thinks a greater number of shaded parts represents a greater fraction

Believes that fractions with the same denominator are equal

Does not understand how to maintain equivalence in fractions

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

Uses numerators to convert fractions to ratio despite different denominators

Does not understand a denominator represents equal parts

Does not recognise when a fraction can be simplified

Cannot identify a fraction on a number line

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

Ordered fractions by comparing denominators rather than using equivalent fractions.

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

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

Removes the common denominator when making equivalent algebraic fractions

When writing fraction from diagram, writes total as denominator

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

Does not find a common denominator when adding/subtracting fractions

Denominator used for other parts rather than total parts

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

When subtracting fractions, subtracts the numerators and denominators

Assumed each part of a fraction diagram is worth 10%

When adding fractions, adds the numerators and denominators

Believes a rectangle has one pair of equal sides

Does not notice all changes in a sequence of visual patterns

Does not understand how to simplify fractions

Identifies the wrong part of a fraction model as the whole

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

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

Assumes the scale is the same when comparing two similar diagrams

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

Thinks that a larger shaded area represents a larger fraction

Finds the shaded section rather than the specified section

Shares equally instead of in given ratio

Does not scale up a ratio to find an equivalent value

Believes that fractions with larger denominators are greater in size

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

Does not recognise the corresponding sides in similar shapes or enlargements

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

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

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

Incorrectly cancels what they believe is a factor in algebraic fractions

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


Question:
### SubjectName: Linear Equations ### ConstructName: Solve linear equations with the variable appearing on both sides, involving negative integers ### Question: Tom and Katie are arguing about a first step to solve this equation:
\(1-4 v=3-9 v\)

Tom says you can write: \( 1+5 v=3 \)

Katie says you can write: \( -4 v=2-9 v \)

Who is correct?

Does not realise that to maintain balance in an equation they must add or subtract the same amount from both sides

Reflects in an axis, instead of the given mirror line

Thinks x = ? is a horizontal line through ?

Translates rather than reflects across a line of symmetry

Does not understand that a point on the line of reflection will stay the same when reflected

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

Thinks y = ? is a vertical line through ?

Assumes every dash on a scale represents an increase of 1

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

Believes that when a shape touches the mirror line, all of its vertices will stay the same when reflected

Struggles to visualise elevations of 3D shapes

Believes a line of reflection must be touching the original shape

When describing a translation, miscounts when crossing an axis

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

Struggles to visualise the new position of an object after transformations

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

Believes that the third vertex of a triangle will share coordinates with both of the other vertices

Struggles to draw 3D shapes on isometric paper

Believes mirror line must lie on a grid line, not in between grid lines

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

Counts squares instead of the correct scale when plotting coordinates

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

Believes point will move diagonally when reflecting in a horizontal or vertical line

Believes that reflection always switches the x and y coordinates

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

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

Thinks the x coordinate of a point on the y axis is 1

Confuses the equations of vertical and horizontal lines

Struggles to visualise 2D triangles in a cuboid

Believes each vertical line on a box plot represents 1 piece of data

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

Uses the same value for both coordinates

Has used the wrong data point on the graph

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

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

Does not realise a triangular prism is a 3D shape

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


Question:
### SubjectName: 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|}{\( 13^{2} \)} \\
\hline\( \ldots \ldots \ldots \) & \( 12^{2} \) \\
\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

Does not notice all changes in a sequence of visual patterns

When asked for a specific term in a sequence just gives the next term

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

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

When asked for a specific term in a sequence gives a different term, which is not the next

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

When given a linear sequence, cannot match it to a visual pattern

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

When asked for the next term in the sequence adds one to the last term given, despite the sequence not going up in ones.

Does not know how to find the next term in a sequence

Uses only the first two terms of a sequence to work out a term-to-term rule

Identifies the term-to-term rule rather than the next term in a sequence

Identifies how much repeated use of the term-to-term rule changes a number but does not apply to the sequence

When asked for the next term in the sequence subtracts one from the last term given, despite the sequence not going down in ones.

Thinks the term in a sequence is double the position value

Identifies the first term rather than the next term in a sequence

Only uses two terms of a sequence to work out a term-to-term rule

Does not know the definition of term or term-to-term rule

Thinks you subtract rather than add when finding the previous term in a descending linear sequence

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

Does not understand that adding on to a multiple can give you the next multiple

Thinks terms in linear sequence are in direct proportion

When asked for a specific term in a sequence, just gives the position number

Goes the wrong direction in the sequence when identifying term-to-term rule

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

Believes there is a multiplicative relationship between values of terms in a linear sequence e.g. 10th term is double 5th term

Believes term to term rule is applied to position number, not previous term

Describes position to term rule using term to term rule

Describes term to term rules using position to term rules

Does not know the meaning of consecutive

Thinks it can't be called a sequence with a repeated number at the start

Thinks the positive and negative terms in a sequence will have the same absolute values

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

Believes to find the next traingle number you add the previous two numbers together i.e. Fibonacci

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

Does not know how to calculate the position of a term in a sequence

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.

Uses term-to-term rule as numerical part of nth term rule

Believes you multiply the constant difference by the term number to get the term value

Believes they can find the nth term based on the first term only

Thinks you add all the bricks in a row in an addition pyramid.

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

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

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: Graphs of Exponentials and Other Powers of x ### ConstructName: Calculate coordinates of points on the graph of y = ax, where x is a positive integer  ### Question: This is a part of the table of values for the equation \( \boldsymbol{y}=\mathbf{9}^{x} \) \begin{tabular}{|c|c|}
\hline\( x \) & \( 0 \) \\
\hline\( y \) & \(\color{gold}\bigstar\) \\
\hline
\end{tabular} What should replace the star?

Forgotten that anything to the power of 0 equals 1

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

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

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

Counts squares instead of the correct scale when plotting coordinates

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

When reading values from graphs, writes down the wrong units

Mixes up the x and y values of the coordinates

When reading graph, chooses nearest marked value on axis instead of reading accurately

Thinks that the scale on axes has to match with the number of squares

Believes a direct proportion graph is a horizontal line

Has used the wrong data point on the graph

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

Believes direct proportion means a 1:1 ratio

Not considered the y value when reading coordinates

Uses the same value for both coordinates

Does not check the labelling of the axes

Multiplies when converting to a larger unit

Believes scales on graphs must match on both axes

Added rather than multiplied by a scale factor

Mixes up the value of two terms when substituting

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

When interpolating using a line of best fit, reads off the value for the closest point to the line rather than the value given by the line of best fit

Assumes the scale is the same when comparing two similar diagrams

Reflects in an axis, instead of the given mirror line

Assumes every dash on a scale represents an increase of 1

Does not understand how to read both x and y coordinates

Thinks x = ? is a horizontal line through ?

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

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

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

Assumes a 1:1 conversion ratio between different currencies

When describing the relationship shown on a scatter graphs, gets the effect on the dependent variable backwards

Shares equally instead of in given ratio

Subtracted rather than divided by a scale factor

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

Does not understand the difference between a value on a graph and the gradient of a graph

Believes that graphs of inverse proportion meet the axes

Mixes up width and height of a rectangle

Confuses the graphs of  direct and inverse proportion

Divides rather than multiplies when scaling up in direct proportion problems

Correct scale but wrong value on protractor reading.

Thinks y = ? is a vertical line through ?

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

Confuses the equations of vertical and horizontal lines

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 Triangles ### ConstructName: Identify a right-angled triangle when presented non-conventionally ### Question: \( A B C D \) is a rectangle.
What type of triangle is triangle \( \mathrm{BCD} \) ? ![Rectangle ABCD with the diagonal shown using a red dotted line]()

Confuses scalene and right angle triangles

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

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

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

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

Rounds to the nearest hundredth, rather than one significant figure

Rounds to the nearest integer rather than one significant figure

Gives a rounded whole number instead of a decimal

Does not understand the value of zeros as placeholders

Counts leading zero as a significant figure

Forgotten placeholders when rounding to significant figures

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

Rounds to the nearest integer when estimating

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

When given an error interval for a decimal that ends in a zero, thinks you give the error interval as if the zero was not present (e.g. 1dp rather than 2dp)

Rounds a decimal instead of using a fraction in division

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

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

Rounds to the wrong degree of accuracy (nearest ten rather than nearest whole number)

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

Identifies first decimal place value when asked for first significant figure

Rounds to the wrong degree of accuracy (1 decimal place rather than nearest whole number)

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

Rounds every number to the same degree instead of rounding to significant figures when estimating

Stops iteration too early when solving equations

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

Changes wrong place value column when rounding

Gives the first significant figure when asked for the second

Removes a zero from within the digits when rounding

Rounds up rather than to one significant figure

Adds or subtracts the degree of accuracy from the original number

Believes they can round when converting units

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

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

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

Removes the decimal point when asked to round a decimal to a whole number

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

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

Rounds the wrong way when interpreting a real-life context

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: Multiplying and Dividing with Decimals ### ConstructName: Divide decimals by 10 ### Question: \( 43.2 \div 10= \)

When dividing a decimal by a multiple of 10, just divides the fractional place values

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

Does not understand that removing an outlier will impact the range

Believes if you add the same value to all numbers in the dataset the range will change

Does not know the definition of the range

When calculating the range does not reorder the data to find the largest number minus the smallest number

Thinks that we cannot calculate the range of negative numbers

Does not know that a reflection would not change the size of the shape

Thinks that we cannot calculate the range if values are repeated

Thinks you cannot find the range if there are an even number of values

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

Thinks the range is just the first value subtracted from the last value, without ordering

Does not include negative numbers when calculating a range

Believes the range is largest add smallest

Thinks that scale factors, other than one, can leave the dimensions of shapes unchanged

Thinks a greater range means more consistent data

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

Does not recognise that an equation will stay balanced if all terms are multiplied/divided by the same amount

Believes when frequencies multiply by a scale factor in a table the median also multiplies by that scale factor

Believes the median is the midpoint of the range

Does not realise that subtracting a larger number will give a smaller answer

subtracts the highest from the lowest when calculating the range

Believes extreme values can affect the median

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

Does not realise we can scale the perimeter as a whole, rather than finding individual side lengths

Assumes the range of a set of data starts at zero

Thinks an enlargement of any scale factor keeps side lengths the same

Believes that we cannot calculate the median of a dataset that contains negative numbers

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

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

Identifies the biggest value for the range

Does not understand that enlargements can make shapes make shapes bigger, smaller and the same size

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

Believes midpoint calculation is just half of the difference

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

Finds the range of possible values of the variable, rather than the range of recorded values

Assumes a fact without considering enough examples

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

Believes interquartile range is the difference between one of the quartiles and the minimum value

Thinks halving is the same as multiplying by 2

Believes when frequencies multiply by a scale factor in a table the mean also multiplies by that scale factor

Thinks a lower average means the data is more consistent

Does not understand that a point on the line of reflection will stay the same when reflected

Believes it's ok to change the scale on an axis for the negative values

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: Simultaneous Equations ### ConstructName: Recognise whether you need to add or subtract linear equations in order to eliminate a variable ### Question: Tom and Katie are discussing this pair of simultaneous equations: \( \begin{array}{l}3 x-2 y=8 \\ 3 x+y=14\end{array} \) Tom says if you add the two equations together you will eliminate a variable

Katie says if you subtract the two equations together you will eliminate a variable

Who is correct?

When solving simultaneous equations thinks that identical terms in given equations can be eliminated by either adding or subtracting.

Rounds up rather than to one significant figure

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

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

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

Believes that rounding must change the number

Rounds down instead of up when next digit is 5

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

Rounds to the nearest integer rather than one significant figure

When rounding up from a 9, does not add one on to the digit to the left

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

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

Rounds the wrong way when interpreting a real-life context

Rounds incorrectly by changing multiple place values

Changes wrong place value column when rounding

Gives a rounded whole number instead of a decimal

Forgotten placeholders when rounding to significant figures

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

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

Believes rounding numbers down would give an overestimate

Rounds every number down when estimating

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

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

Does not know how to round up when a decimal place value column contains a 9

Rounds to the nearest integer when estimating

Does not understand you can round to numbers other than powers of 10

Believes that when finding bounds you round to the nearest whole number

Believes rounding to 1 decimal pleace means rounding to the nearest integer, regardless of the value of the number

Rounds every number to the same degree instead of rounding to significant figures when estimating

Mixes up greater than and less than symbols

Removes the decimal point when asked to round a decimal to a whole number

Rounds to the nearest hundredth, rather than one significant figure

Removes a zero from within the digits when rounding

Counts leading zero as a significant figure

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

Rounds incorrectly when the nearest 10 is a multiple of 100

Cannot evaluate if an estimate is an underestimate or overestimate

Believes they can round when converting units

Adds or subtracts the degree of accuracy from the original number

Gives the first significant figure when asked for the second

Confuses place value when rounding to estimate

When using interpolation to estimate frequency that is above a certain value, believes they can add the estimated frequency up to that value in its class, rather than the estimated frequency after it

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