Error analysis - Methodical analysis of task performance

Performance analysis and methodical recommendations

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EMMentor_Algebra

This page demonstrates how one-task performance is analysed. Shown are the results of step by step analysis of errors. You can see a summary of errors for each solution step. This option is only available in the program EMMentor. EMMentor shows you why you are struggling in math and how to improve. At the bottom of this page we offer you to try examples of learning techniques available in the programs EMSolution and EMMentor.

Multifactor error analysis and methodical recommendations

Methodical recommendations are based on a joint multifactor analysis of a set of performed tasks. Multifactor analysis means that user's errors are analysed with regard to multiple parameters - to formulas, definitions and rules used in the performed tasks, to applied techniques, topics and types of tasks. Such comprehensive analysis enables EMMentor to detect vacancies in user's knowledge and skills and generate an optimal set of tasks to close the revealed gaps. EMMentor helps you find out why you are struggling in math and provides a qualified guidance in training problem solving skills.

Journal

The Journal keeps a record of performed tasks with grades (marks) and summaries of errors and activates the results of multifactor error analysis with concluding methodical recommendations.

Methodical analysis of task performance

23.2.2005

3:4:39

Educational task:

	Rational equations
	Quadratic equations
	Equations. Method of substitution. Grouping method
	Solve the equation

{(- 40 y^{2} + 26 y - 3)}^{2} + {(20 y^{2} - 13 y + 6)}^{2} = {(- 20 y^{2} + 13 y - 2)}^{2} + 2 \cdot (20 y^{2} - 13 y + 6) \cdot (- 40 y^{2} + 26 y - 3)

Methodical task:

	To relate objectives of the steps to the steps of solution
	To relate formulations to the steps of solution
	To relate formulations to the steps of solution
	To relate formulas to the steps of solution

The task was discharged by:	xxx yyy zzz
Affiliation:	xxx yyy zzz
Number of errors made:	22
Number of used hints:	32
Mark/grade (USA):	E 1

The task is not accomplished

Too much errors. Try a simpler educational task. Select a task from the listed groups of methods:

	Scheme "To relate categories with the step 6"
	Scheme "To relate categories with the step 5"
	Scheme "To relate categories with the step 4"

Methodical analysis of task performance

	Analysis of solution steps
	Analysis of math categories

The course of solution was either wrong or nonoptimal in the selection of the following transformations:

0. Solve the equation

1. The considered expression contains no forbidden operations (division by zero, extraction of the even roots from negative numbers, etc.). Therefore, the domain of definition is:

2. Considering that there are only identical elementary functions of the argument, let us replace these functions by the rule: f(x)=t, where x is the old argument, t - the new argument, and f - the replaceable function

3. Using the commutative law of addition, let us group the addends in such a way that the formula of perfect square can be applied

4. Let us transform the expression by applying the formula of perfect square

5. Let's combine polynomials, using the definition of operation of addition

6. Let's collect similar terms of polynomial

7. Let's add up coefficients at the like terms

9. Let's move expressions from one side of equation to the other

10. Let's combine polynomials, using the definition of operation of addition

11. Let's collect similar terms of polynomial

12. Let's add up coefficients at the like terms

14. Let's make the inverse change of the variable

15. Using properties of product, let us replace the equation(s) by the aggregation of equations

16. Using properties of equations, let us solve the set of equations

17. Answer

Educational task:

0. Solve the equation

{(- 40 y^{2} + 26 y - 3)}^{2} + {(20 y^{2} - 13 y + 6)}^{2} = {(- 20 y^{2} + 13 y - 2)}^{2} + 2 \cdot (20 y^{2} - 13 y + 6) \cdot (- 40 y^{2} + 26 y - 3)

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	2	0
To relate formulations to the steps of solution	0	0
To relate formulations to the steps of solution	0	0
To relate formulas to the steps of solution	0	0

1. The considered expression contains no forbidden operations (division by zero, extraction of the even roots from negative numbers, etc.). Therefore, the domain of definition is:

y \in R

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	0
To relate formulations to the steps of solution	0	0
To relate formulations to the steps of solution	0	0
To relate formulas to the steps of solution	1	0

{(- 40 u - 3)}^{2} + {(20 u + 6)}^{2} = {(- 20 u - 2)}^{2} + 2 \cdot (20 u + 6) \cdot (- 40 u - 3)

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	2	0
To relate formulations to the steps of solution	0	0
To relate formulations to the steps of solution	0	0
To relate formulas to the steps of solution	2	0

3. Using the commutative law of addition, let us group the addends in such a way that the formula of perfect square can be applied

{(- 40 u - 3)}^{2} - 2 \cdot (20 u + 6) \cdot (- 40 u - 3) + {(20 u + 6)}^{2} = {(- 20 u - 2)}^{2}

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	0
To relate formulations to the steps of solution	1	0
To relate formulations to the steps of solution	0	0
To relate formulas to the steps of solution	0	0

4. Let us transform the expression by applying the formula of perfect square

{((- 40 u - 3) - (20 u + 6))}^{2} = {(- 20 u - 2)}^{2}

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	1	0
To relate formulations to the steps of solution	1	1
To relate formulations to the steps of solution	0	1
To relate formulas to the steps of solution	0	1

5. Let's combine polynomials, using the definition of operation of addition

{(- 40 u - 3 - 20 u - 6)}^{2} = {(- 20 u - 2)}^{2}

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulations to the steps of solution	2	0
To relate formulas to the steps of solution	1	0

6. Let's collect similar terms of polynomial

{(- 40 u - 20 u - 3 - 6)}^{2} = {(- 20 u - 2)}^{2}

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	0
To relate formulations to the steps of solution	0	0
To relate formulations to the steps of solution	0	0
To relate formulas to the steps of solution	1	0

7. Let's add up coefficients at the like terms

{(- 60 u - 9)}^{2} = {(- 20 u - 2)}^{2}

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	2	0
To relate formulations to the steps of solution	0	0
To relate formulations to the steps of solution	0	0
To relate formulas to the steps of solution	1	0

8. Let's raise polynomial(s) to natural power

(3600 u^{2} + 1080 u + 81) = (400 u^{2} + 80 u + 4)

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	0
To relate formulations to the steps of solution	0	0
To relate formulations to the steps of solution	0	0
To relate formulas to the steps of solution	0	0

9. Let's move expressions from one side of equation to the other

(3600 u^{2} + 1080 u + 81) - (400 u^{2} + 80 u + 4) = 0

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	2	0
To relate formulations to the steps of solution	0	0
To relate formulations to the steps of solution	0	0
To relate formulas to the steps of solution	2	0

10. Let's combine polynomials, using the definition of operation of addition

(3600 u^{2} + 1080 u + 81 - 400 u^{2} - 80 u - 4) = 0

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulas to the steps of solution	0	1

11. Let's collect similar terms of polynomial

3600 u^{2} - 400 u^{2} + 1080 u - 80 u + 81 - 4 = 0

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulas to the steps of solution	0	1

12. Let's add up coefficients at the like terms

3200 u^{2} + 1000 u + 77 = 0

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulas to the steps of solution	1	0

13. Let's factor quadratic trinomial(s) if possible

3200 \cdot (u + \frac{7}{40}) \cdot (u + \frac{11}{80}) = 0

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	0
To relate formulations to the steps of solution	0	0
To relate formulations to the steps of solution	0	0
To relate formulas to the steps of solution	0	0

14. Let's make the inverse change of the variable

3200 \cdot (y^{2} - \frac{13}{20} y + \frac{7}{40}) \cdot (y^{2} - \frac{13}{20} y + \frac{11}{80}) = 0

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulas to the steps of solution	0	1

15. Using properties of product, let us replace the equation(s) by the aggregation of equations

[\begin{matrix} y^{2} - \frac{13}{20} y + \frac{7}{40} = 0 \\ y^{2} - \frac{13}{20} y + \frac{11}{80} = 0 \end{matrix}

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulas to the steps of solution	0	1

16. Using properties of equations, let us solve the set of equations

y \in \emptyset

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulas to the steps of solution	0	1

17. Answer

y \in \emptyset

Methodical task:	Errors:	Hints:	C
To relate objectives of the steps to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulations to the steps of solution	0	1
To relate formulas to the steps of solution	0	1

Analysis of math categories

	Objective of the step
	Definition
	Formulation
	Formula

Objective of the step

Errors:	Hints:	Mathematical categories
2	0	To solve an equation means to find all its roots or to prove that there is none
1	0	To apply a formula means to transform an expression using the formula's relation, with allowance made for the region of formula definition
0	1	To give an answer means to find and write down the solution of equation
0	2	To add up polynomials means to transform an expression using the definition of operation of addition and definition of polynomial
2	0	To make a change of function means to replace the function f(x) by the new argument t according to the rule f(x)=t, where x is the old argument, t - the new argument, f (x) - the replaceable function, and to indicate the region of function f variation
0	1	To make the inverse replacement of function means to replace the argument by the function according to the rule: t=f(x), where the function and argument were defined at the forward replacement
0	1	To collect similar terms of polynomial means to group together similar monomials, using the commutative law
2	1	To add up coefficients at the similar terms of polynomial means to add up, using the distributive law, the numerical coefficients at the similar monomials
2	0	To move expressions from one side of equation to the other means to set up a new equation equivalent to the given one, using the addition principle of equations equivalence
0	1	To replace an equation by the aggregation of equations means to use the theorems of equivalence of equations and the definition of equation
0	1	To solve a set of equations means to find all their roots or to prove that there is none

Definition

Errors:	Hints:	Mathematical categories
0	1	Two equations are called equivalent if their roots are equal and domains of definition coincide
0	1	The addition principle (property) of equivalence of equations: The same expression may be added to or subtracted from both sides of equation
0	1	An inequality is referred to as an algebraic expression containing logical connectives greater than (greater than or equal to) or less than (less than or equal to), and an unknown
0	1	Polynomial is referred to as an algebraic sum of monomials
0	1	A function is referred to as a rule by which each value of argument is associated with a unique value of function
0	1	Monomials are called the like terms of a polynomial if they differ by the value of a numerical constant only
1	1	The formula of perfect square: The square of sum (difference) of two numbers equals to the square of the first number plus (minus) the doubled product of the first by the second number plus the square of the second number
1	0	The grouping method consists in rearrangement of addends in the order suitable for the subsequent transformations
0	2	An operation of addition is referred to as a binary operation on a set, satisfying the associative and commutative laws with the existing neutral (zero point) element and inverse elements

Formulation

Errors:	Hints:	Mathematical categories
0	1	A product equals to zero when at least one multiplier equals to zero and the other multipliers do not lose their numeric sense
0	1	A solution of an inequality is referred to as a range of unknown under which an inequality is valid
0	1	Monomial is referred to as an algebraic expression consisting of numerals, variables to different natural powers and operation of multiplication
0	1	A region of formula definition is referred to as all values of argument and parameters entering the formula, at which the formula's relation is defined
0	1	A substitution is referred to as a replacement of argument by a function of new argument according to the rule: x=f(t) (where x is the old argument, t - the new argument, f - the function of substitution), supplemented with indication of the region of substitution definition
0	1	The addition principle of equivalence of equations: The same expression may be added to and subtracted from both sides of equation
2	1	A binary operation on a set is referred to as an operation relating (transforming) two elements of the set to the third element of this set: "aTb=c", where a, b, c are the elements of the set, and T is a binary operation
0	1	An operation of addition is referred to as a binary operation on a set, satisfying the associative and commutative laws with the existing neutral (zero point) element and inverse elements

Formula

Errors:	Hints:	Mathematical categories
1	1	$a + b = b + a$
0	1	${(a \pm b)}^{2} = a^{2} \pm 2 a b + b^{2}$
2	0	$P + a = Q <=> P = Q - a$
0	1	$k x + b = 0 <=> x = - k^{-1} b$
2	0	$x^{2} + b x + c = t$
0	1	$x = f (t) => t = f^{-1} (x)$
1	1	$a + b = c$
0	1	$\prod_{k = 0}^{n} f_{k} (x) = 0 \Rightarrow [\begin{array}{l} f_{k} (x) = 0 \\ k = 0,1,... n \end{array}$
1	0	$f (x) \geq a x \in Ω \cap D (f)$
0	1	$f (x) = 0 \exists Ω \subset R : {\begin{cases} \forall λ \in Ω f (λ) \equiv 0 \\ \forall λ \notin Ω f (λ) \neq 0 \end{cases} \Rightarrow x \in Ω$
2	0	$a \cdot (b + c) = a \cdot b + a \cdot c$

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