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

3:8:22

Educational task:

	Rational equations
	Fractional algebraic equations
	Fractional algebraic equations reduced to linear equations
	Solve the equation

\frac{3 t + 4}{2 t - 3} + \frac{4 t - 3}{- 4 t - 3} = \frac{1}{2}

Methodical task:

To select the objective of the next step of transformation

The task was discharged by:	xxx yyy zzz
Affiliation:	xxx yyy zzz
Number of errors made:	14
Number of used hints:	9
Mark/grade (USA):	E 32

The task is not accomplished

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

	Scheme "To apply categories 4"
	Scheme "To apply categories 3"
	Scheme "To apply categories 2"

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. Let's find the domain of definition D(f) for the considered expression, taking into account that the division by zero is not defined

2. Let's solve linear equation(s), applying the addition and multiplication principles of equations equivalence

4. Using the distributive law and the rule of multiplication of polynomials "each by each", let's multiply polynomials in the numerator of the fraction

6. Let's collect similar terms of polynomial

7. Let's add up similar terms of polynomial

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

9. Let's add up fractions applying the rule of addition of fractions

10. Using the distributive law and the rule of multiplication of polynomials "each by each", let's multiply polynomials in the numerator of the fraction

11. Using the distributive law and the rule of multiplication of polynomials "each by each", let's multiply polynomials in the numerator of the fraction

12. Let's remove brackets, using the distributive law

15. Let's clear the fraction, using the multiplication principle of equivalence of equations (inequalities)

16. Let's multiply both sides of equation by "-1"

17. Using the addition principle of equivalence of equations, let us move the numeric addend from the left-hand side of the equation to its right-hand side

18. Using the multiplication principle of equivalence of equations, let us divide both sides of the equation by numerical coefficient at the argument

19. Let us account for the domain of definition

Educational task:

0. Solve the equation

\frac{3 t + 4}{2 t - 3} + \frac{4 t - 3}{- 4 t - 3} = \frac{1}{2}

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	2	0

1. Let's find the domain of definition D(f) for the considered expression, taking into account that the division by zero is not defined

{\begin{matrix} 2 t - 3 \neq 0 \\ - 4 t - 3 \neq 0 \end{matrix}

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	2	0

2. Let's solve linear equation(s), applying the addition and multiplication principles of equations equivalence

{\begin{matrix} t \neq \frac{3}{2} \\ t \neq - \frac{3}{4} \end{matrix}

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	2	0

3. Let's add up fractions applying the rule of addition of fractions

\frac{(3 t + 4) \cdot (- 4 t - 3) + (4 t - 3) \cdot (2 t - 3)}{(2 t - 3) \cdot (- 4 t - 3)} = \frac{1}{2}

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	0

4. Using the distributive law and the rule of multiplication of polynomials "each by each", let's multiply polynomials in the numerator of the fraction

\frac{(- 12 t^{2} - 9 t - 16 t - 12) + (8 t^{2} - 12 t - 6 t + 9)}{(2 t - 3) \cdot (- 4 t - 3)} = \frac{1}{2}

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	1	0

5. Let's remove brackets, using the distributive law

\frac{- 12 t^{2} - 9 t - 16 t - 12 + 8 t^{2} - 12 t - 6 t + 9}{(2 t - 3) \cdot (- 4 t - 3)} = \frac{1}{2}

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	0

6. Let's collect similar terms of polynomial

\frac{- 12 t^{2} + 8 t^{2} - 9 t - 16 t - 12 t - 6 t - 12 + 9}{(2 t - 3) \cdot (- 4 t - 3)} = \frac{1}{2}

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	2	0

7. Let's add up similar terms of polynomial

\frac{- 4 t^{2} - 43 t - 3}{(2 t - 3) \cdot (- 4 t - 3)} = \frac{1}{2}

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	1

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

\frac{- 4 t^{2} - 43 t - 3}{(2 t - 3) \cdot (- 4 t - 3)} - (\frac{1}{2}) = 0

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	1

9. Let's add up fractions applying the rule of addition of fractions

\frac{- 4 t^{2} - 43 t - 3 - (\frac{1}{2}) \cdot (2 t - 3) \cdot (- 4 t - 3)}{(2 t - 3) \cdot (- 4 t - 3)} = 0

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	1

10. Using the distributive law and the rule of multiplication of polynomials "each by each", let's multiply polynomials in the numerator of the fraction

\frac{- 4 t^{2} - 43 t - 3 - (t - \frac{3}{2}) \cdot (- 4 t - 3)}{(2 t - 3) \cdot (- 4 t - 3)} = 0

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	1

11. Using the distributive law and the rule of multiplication of polynomials "each by each", let's multiply polynomials in the numerator of the fraction

\frac{- 4 t^{2} - 43 t - 3 - (- 4 t^{2} - 3 t + 6 t + \frac{9}{2})}{(2 t - 3) \cdot (- 4 t - 3)} = 0

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	1

12. Let's remove brackets, using the distributive law

\frac{- 4 t^{2} - 43 t - 3 + 4 t^{2} + 3 t - 6 t - \frac{9}{2}}{(2 t - 3) \cdot (- 4 t - 3)} = 0

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	1

13. Let's collect similar terms of polynomial

\frac{- 4 t^{2} + 4 t^{2} - 43 t + 3 t - 6 t - 3 - \frac{9}{2}}{(2 t - 3) \cdot (- 4 t - 3)} = 0

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	0

14. Let's add up similar terms of polynomial

\frac{- 46 t - \frac{15}{2}}{(2 t - 3) \cdot (- 4 t - 3)} = 0

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	0

15. Let's clear the fraction, using the multiplication principle of equivalence of equations (inequalities)

- 46 t - \frac{15}{2} = 0

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	1	1

16. Let's multiply both sides of equation by "-1"

46 t + \frac{15}{2} = 0

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	1

17. Using the addition principle of equivalence of equations, let us move the numeric addend from the left-hand side of the equation to its right-hand side

46 t = - \frac{15}{2}

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	1

18. Using the multiplication principle of equivalence of equations, let us divide both sides of the equation by numerical coefficient at the argument

t = - \frac{15}{92}

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	2	0

19. Let us account for the domain of definition

t = - \frac{15}{92} \in D(f)

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	2	0

20. Answer

\forall t \in {- \frac{15}{92}}

Methodical task:	Errors:	Hints:	C
To select the objective of the next step of transformation	0	0

Analysis of math categories

Objective of the step

Errors:	Hints:	Mathematical categories
6	0	To solve an equation means to find all its roots or to prove that there is none
2	0	To find the domain of definition for an expression means to write a system of inequalities determining the forbidden values of argument at which the expression is undefined
2	0	To account for the domain of definition means to remove all forbidden values from the solution
0	1	To add up fractions means to make up a new fraction, using the rule of addition of fractions
1	1	To clear a fraction means to multiply both sides of an equation (inequality) by the expression equal to the common nonzero denominator, resting on the multiplication principle of equivalence of equations (inequalities)
0	1	To remove brackets means to delete the characters of algebraic brackets from the notation of expression according to the rule induced by the distributive law
1	2	To multiply polynomials means to transform an expression, using the rule of multiplication of polynomials "each by each" (a conclusion of the distributive and associative laws of multiplication)
0	1	To move an addend from one side of an equation to the other means to transform an equation applying the addition principle of equivalence of equations
2	0	To collect similar terms of polynomial means to group together similar monomials, using the commutative law
0	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
0	1	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 multiply both sides of equation by "-1" means to set up a new equation equivalent to the given one

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