Error analysis - Methodical analysis of task performance

Performance analysis and methodical recommendations

This page was created by

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

2:59:21

Educational task:

	Rational equations
	Quadratic equations
	Equations. Grouping method
	Solve the equation

{(2 t + 9)}^{2} + 2 \cdot (2 t + 9) \cdot (5 t + 4) + {(5 t + 4)}^{2} = (- 4 t - 6)

Methodical task:

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

The task was discharged by:	xxx yyy zzz
Affiliation:	xxx yyy zzz
Number of errors made:	27
Number of used hints:	9
Mark/grade (USA):	E 19

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 the step with category 6"
	Scheme "To relate the step with category 5"
	Scheme "To relate the step with category 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. Let us transform the expression by applying the formula of perfect square

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

3. Let's collect similar terms of polynomial

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

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

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

8. Let's collect similar terms of polynomial

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

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

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

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

13. Answer

Educational task:

0. Solve the equation

{(2 t + 9)}^{2} + 2 \cdot (2 t + 9) \cdot (5 t + 4) + {(5 t + 4)}^{2} = (- 4 t - 6)

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

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

{((2 t + 9) + (5 t + 4))}^{2} = (- 4 t - 6)

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

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

{(2 t + 9 + 5 t + 4)}^{2} = (- 4 t - 6)

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

3. Let's collect similar terms of polynomial

{(2 t + 5 t + 9 + 4)}^{2} = (- 4 t - 6)

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

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

{(7 t + 13)}^{2} = (- 4 t - 6)

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

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

(49 t^{2} + 182 t + 169) = (- 4 t - 6)

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

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

(49 t^{2} + 182 t + 169) - (- 4 t - 6) = 0

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

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

(49 t^{2} + 182 t + 169 + 4 t + 6) = 0

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

8. Let's collect similar terms of polynomial

49 t^{2} + 182 t + 4 t + 169 + 6 = 0

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

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

49 t^{2} + 186 t + 175 = 0

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

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

49 \cdot (t + \frac{93}{49} + \sqrt[2]{\frac{74}{2401}}) \cdot (t + \frac{93}{49} - \sqrt[2]{\frac{74}{2401}}) = 0

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

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

[\begin{matrix} t + \frac{93}{49} + \sqrt[2]{\frac{74}{2401}} = 0 \\ t + \frac{93}{49} - \sqrt[2]{\frac{74}{2401}} = 0 \end{matrix}

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

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

[\begin{matrix} t = - \frac{93}{49} - \sqrt[2]{\frac{74}{2401}} \\ t = - \frac{93}{49} + \sqrt[2]{\frac{74}{2401}} \end{matrix}

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

13. Answer

\forall t \in {- \frac{93}{49} - \sqrt[2]{\frac{74}{2401}}; - \frac{93}{49} + \sqrt[2]{\frac{74}{2401}}}

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

Analysis of math categories

	Objective of the step
	Definition
	Formulation
	Formula

Objective of the step

Errors:	Hints:	Mathematical categories
1	0	To solve an equation means to find all its roots or to prove that there is none
1	0	To raise a polynomial to natural power means, in general case, to apply the binomial formula
0	1	To give an answer means to find and write down the solution of equation
1	0	To add up polynomials means to transform an expression using the definition of operation of addition and definition of polynomial
3	0	To collect similar terms of polynomial means to group together similar monomials, using the commutative law
2	0	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
1	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
2	0	To replace an equation by the aggregation of equations means to use the theorems of equivalence of equations and the definition of equation
1	0	To factor a quadratic trinomial means to present it as a product of linear expressions

Definition

Errors:	Hints:	Mathematical categories
1	1	An equation is referred to as an algebraic expression containing the symbol of equality and an unknown
1	0	Polynomial is referred to as an algebraic sum of monomials
1	1	The Newton binomial formula is referred to as a formula allowing to raise expressions, consisting of two or more addends, to natural powers. When the power exponent equals to 2 or 3, the binomial formula has the special name of square (or cube) of sum (difference)

Formulation

Errors:	Hints:	Mathematical categories
0	1	A root of equation is referred to as a value of the unknown converting the equation into identity
2	0	Monomial is referred to as an algebraic expression consisting of numerals, variables to different natural powers and operation of multiplication
0	1	The binomial coefficients are referred to as factors entering the binomial formula; they depend on power exponent. These factors are calculated using the combinatorial relations
1	0	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	The addition principle of equivalence of equations: The same expression may be added to and subtracted from both sides of equation
3	0	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

Formula

Errors:	Hints:	Mathematical categories
3	0	$a + b = b + a$
1	0	$P + a = Q <=> P = Q - a$
0	1	$k x + b = 0 <=> x = - k^{-1} b$
2	0	$a x^{2} + b x + c = a \cdot (x - \frac{- b + \sqrt[2]{b^{2} - 4 a c}}{2 a}) \cdot (x - \frac{- b - \sqrt[2]{b^{2} - 4 a c}}{2 a})$
0	1	$\begin{array}{l} {(a + b)}^{n} = \sum_{k = 0}^{n} C_{n}^{k} \cdot a^{(n - k)} \cdot b^{k}, \\ {\begin{cases} C_{n}^{k} = \frac{n!}{(n - k)! \cdot k!} \\ 0! = 1 n! = (n - 1)! \cdot n \end{cases} \\ E x a m p l e 1 : {(a \pm b)}^{5} = C_{5}^{0} \cdot a^{5} \pm C_{5}^{1} \cdot a^{4} \cdot b + C_{5}^{2} \cdot a^{3} \cdot b^{2} \pm C_{5}^{3} \cdot a^{2} \cdot b^{3} + C_{5}^{4} \cdot a \cdot b^{4} \pm C_{5}^{5} \cdot b^{5} \\ {(a \pm b)}^{5} = a^{5} \pm 5 \cdot a^{4} \cdot b + 10 \cdot a^{3} \cdot b^{2} \pm 10 \cdot a^{2} \cdot b^{3} + 5 \cdot a \cdot b^{4} \pm b^{5} \\ E x a m p l e 2 : C_{7}^{3} = \frac{7!}{(7 - 3)! \cdot 3!} = \frac{4! \cdot 5 \cdot 6 \cdot 7}{4! \cdot 3!} = \frac{5 \cdot 6 \cdot 7}{1 \cdot 2 \cdot 3} = 35 \end{array}$
0	1	$f (x) = 0 \exists Ω \subset R : {\begin{cases} \forall λ \in Ω f (λ) \equiv 0 \\ \forall λ \notin Ω f (λ) \neq 0 \end{cases} \Rightarrow x \in Ω$

More...

Utrecht, The Netherlands