### Symbolic Checking Options

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These options are grouped in ways to help guide the evaluation of the student response. As a general practice, you should use the PREVIEW mode to check your questions with examples that are correct and also incorrect to check the grading is working in the way you intend it to. In most cases, the Edulastic auto-grader will identify the correct context, but in some cases there is ambiguity, and so you will need to manually adjust the grading process using these options.

See some examples here.

The symbolic evaluation considers mathematical equivalence and specific rules that are applied to the student's entry.

Symbolic evaluation works with expressions with variables, e.g. x2y ,or  equations, such as log(x) - y = 0 and values such as 11.5, 99.5%, 1/2.

By example, consider a question that has a response of

x+2

There are many mathematically equivalent forms of this expression such as:  2 + x, 1 + x + 1, 2 + 1*x.

If just a subset of responses is required from the student, such as specifically "x+2" then either use the Literal evaluation mode or customize the Symbolic checking options as described below.

Here is another example of mathematical equivalence when the expected answer is the number 12

 Student Response Expected Response Result 12 12 true 12.0 12 true 3 x 4 12 true 120 / 10 12 true 12 apples 12 false √(144) 12 true

The equivalences described in this document are scored as true/false and the corresponding grading of the question will be correct/incorrect or 1/0 points.

Use the options below for finer control which of the above student inputs are considered correct.

Note the difference with Literal checking in which most of the above student inputs would be scored as false.

Note that in general variables can be anything, e.g. q, x1 or abc or xyz,  but we recommend as a safe practice - especially when advanced options are selected, that you use variables names such as w, x, y, z in expressions to avoid possible confusion with units such as "s" which the evaluator might consider as the units "seconds" and then interpret it differently depending on which context it assumes. Other common units are m for meters and g for grams.

The symbolic mode options can be assessed by clicking on the green "CHANGE" button below the Expected Answer field: Note that you can hover over the little "i" symbols to get more information on the options.

There are three sections of options:

• MISCELLANEOUS

Symbolic Checking Rules:

Decimal: numeric entries with a decimal point followed by some number of digits, e.g.  4.36

Scientific: m × 10n,  e.g. 4.23E-2

Simplified: generally, this means to make less complicated, clearer, or easier. Cancel common factors and group terms. This can be used for fractions or expressions.

Factored: split into a product of simpler expressions. The opposite is Expanded.

Improper Fraction: the numerator is bigger than the denominator, e.g. 9/7  Only use this option for purely numeric fractions, e.g. the evaluation may not work as you expect if the answer is "11x/2".

Rationalized: No roots (e.g. a square root or cube root) in the denominator

Mixed Fraction: A whole number and a proper fraction, e.g. 1 1/2. Only use this option for purely numeric fractions, e.g. the evaluation may not work as you expect if the answer is "7 x/2" which contains a variable x. On the other hand, 7 1/2 would be handled properly.

Exponential Form: represent repeated multiplications of the same number by writing the number as a base with the number of repeats, e.g. 24, x3

Rational: No roots in the expression.

Note that these forms can be entered in the keypads with specific notations, e.g. the mixed fraction

1 2/3 (one and two-thirds) can be entered as follows.

Use the arrow, up and down, and tab keys to navigate the notation structure. Here is how it looks with the 1 and 2 entered for the mixed fraction... The grey box is where the three should be entered: To navigate between the different areas use the arrows and tab keys on the device keyboard: More Examples of these rules are given below.

Simplified:

For example, 3/3 and 4/2 are not simplified, while 4 and ½ are simplified. This requirement can be applied to expressions and purely numeric enters.

 Student Response Result 4/4 false 3x+y true x+x+x+y false 6x+2y false x2+2x+1 true (x+1)2 false (x+1)(x+2) false

Please refer to the "factored" and "expanded" option below if you wish to change the grading logic.

This option also handles equations of the form f(x,y)=0

1. grouping of terms like "x+x" vs "2x"
2. removing denominators in fractions
3. removing common factors
4. the first term in polynomial expression is positive.
 Student Response Result Correct Version x+y=0 true 3x+y = 0 true x+x+x+y = 0 false 3x + y = 0 1/2 x + y =0 false x + 2y = 0 -x^2 +x +7 = 0 false x^2 - x - 7 =0 2 a + 4 b = 0 false a + 2 b = 0

Expanded

Checks if an expression is in its expanded form or not. A factorized expression is not in its expanded form.

 Student Response Result 2x true x2+3x true (x+3)(x+y) false

The last row would be considered as Factored.

Factored

Checks if an expression is in its expanded form or not. A factorized expression is not in its expanded form.

 Student Response Result 2y(y-4)(y-6) true x2+3x false (x+3)(x+y) true (x+1)2 true

The middle row would be considered as Expanded.

Improper Fraction

The numerator is larger than the denominator, e.g. 11/2

 Student Response Result 7/2 true 3/4 false 7x/4 ?

For the bottom row, note that you should use this option for purely numeric fractions, as expressions with variables might not evaluate properly because the variable value could change the form of the entry.

Mixed Fraction

This is defined as having an integer followed by a proper fraction. For example, 1 ½ is a mixed fraction, while 3/2 is not.

 Student Response Result 1 3/4 true 7/4 false 7x/4 ?

For the bottom row, note that you should use this option for purely numeric fractions, and expressions with variables may not evaluate properly.

Decimal:

 Student Response Result 10.0 true 3/2 false

Decimal and Two Significant Decimal Places:

 Student Response Result 7.1 false 7.11 true

Scientific and Two Significant Decimal Places:

Consider the following responses for a raw value of 711:

 Student Response Result 7.1E+2 false 7.11E+2 true 7.11e2 true 7.11 x 102 true

Exponential Form

 Student Response Result 23 true 2x2x2 false x2 true

Special Handling:

Interpret e as Euler's Number:

This will consider the letter ‘e’ or ‘\exp’ to be the same as Napier’s constant of 2.718...

 Student Response Expected Result \ln ex = 1 x = 1 true ex - 1 \exp(x) - 1 true

Interpret as an ordered list

A sequence of more than three items is considered to be a list. We can use this option to enforce the checking of the student's response as a list form as opposed to a set form (see below). In a list, the order is important, so for example 2, 1, 0 will not be the same as 0, 1, 2. Lists can also include variables and expressions. Examples of Lists are as follows:

 Student Response Expected Result 1,2 1,2,3 false 0,1 0,1 true 1,0 0,1 false 4,654 (4,654) true

Interpret as a set

Use this to analyze the student’s response is a set. In a set, the order is not important, so for example 2,1 will be the same as {1,2}.  To force the interpretation to be a set, use the advanced option as follows: Sets and Lists can also include variables and expressions.

 Student Response Expected Result (3,2,4) {2,3,4} true 1,2 1,2,3 false

Note that the first case would be false in the case of a List since the order is different.

The following forms are automatically inferred by the evaluation system based on the typographic form of bracket notations that are provided in the Expected Answer entry:

{ } ⇒ set

[ ] ⇒ List

(a,b) ⇒ Interval

(a,b] ⇒ Interval, but if b<a then an error will be returned

[a,b) ⇒ Interval, but if b<a then an error will be returned

<0,1> ⇒ Vector

But if the advanced options as mentioned above such as "interpret as a list" or "interpret as a set" is provided then these will override the typographic form identification that is provided.

Note that the parenthesis form of the student entry does not necessarily have to match the form of the expected answer. If your intention is to require the student to enter the specific parenthesis for the form, then you may need to set the specific options as discussed here. The reason for this behavior is that evaluation system is designed to handle the majority of possible cases by default in an intelligent way and so teachers would not need to worry about the parenthesis formats.

Examples:

 Option Student Response Expected Result interpret as a set 3,2,4 [2,3,4] true interpret as a set 1,3 [1,2,3] false none 2,1 (1,2) false

The last case (1,2) will be interpreted as an interval, or a coordinate point in the x-y plane. For intervals, they can have smooth braces or square if the ending value is included. For example, (0,2] as an interval includes all Real numbers between zero and 2, including the value 2. Zero is not included.

More examples of default interpretation of the Expected Entry are as follows:

 Option Student Response Expected Result none (0,1) [0,1] true none (1,0] (1,0] Error none [1,2) (1,2] false

In the first case, the student input can be a set, list, or an ordered pair. If an interval is intended then use the "Interval Notation" option In the second case, the form of the expected input is presumed to be an interval since the parenthesis do not match, but the second coordinate is less than the first, an error will be returned

In the third case, the form of the expected entry is presumed to be an interval, but the student entry does not match it, so the result is false.

Note that there are other uses for these forms:

Braces, .e.g.  { }  can also be used for grouping expressions in LaTex

Parenthesis, e.g. ( ) are also for coordinate pairs or intervals less than or greater than.

Brackets, .e.g. [ ] are for lists and inclusive interval endpoints

Compare both sides of an inequality

Use this when the answer involves equality or inequality and both sides need to be checked. This means the Left-hand side of the student response is compared with the Left-hand side of the expected, and the right-hand side of the student response is compared with the Right-hand side of the expected.
For example if the expected answer is: 5+1 = 6 and the student enters 4+2 = 3 + 3, this will return true, while student responses of 5+2 = 6 or 5+2 =7 will both return false.

 Student Response Expected Result x +1 = 2y + 4 x +1 = 2y + 4 true x = 2y + 3 x +1 = 2y + 4 false

In the bottom row the equation has been simplified, but the LHS and RHS of the student entry does not match the expected.

This option is particularly important for TEMPLATE questions, e.g. Another way that you could try to implement these kinds of questions is shown in equation response questions.  Here is an another use case for this compare sides option:

Students are learning to write algebraic equations, and you want them to isolate the variable in a certain way...

So x+10=15 or 10+x=15 are ok but  15-10=x should be graded as wrong.

There is one special case of this option. If the expected answer has an inequality, e.g. x < y and the student enters y > x, then it will still be graded as true because the inequalities will be aligned.

 Student Response Expected Result x < y+ 1 x < y + 1 true y +1 < x x < y + 1 true

Other options:

Set Decimal Separator. The default is dot, but it can be changed to comma.

Set Thousands Separator. The default is comma, but it can be changed to comma, dot or space.

Restrict to Numeric Input. This will only allow the students to enter numbers, include. dot, +, -, /  for their responses

Use Degrees instead of Radians. For trigonometry, the default is to assume the argument is radians, e.g. sin(\pi/2) = 1. If you select this option, then sin(90) =1

Restrict Variables used to. Enter x,y means the student can only use the characters x or y in their expressions.

Allow Tolerance. This can be a percentage, e.g. 10%, or an absolute amount. For example if tolerance = 2 and the expected answer is 10, then 8,9,10,11,12 would all be considered as correct answers.

Significant Decimal Places. The student needs to enter this many decimal places. For example, if it is set to two, and the student enters 10.1 then this will be graded as incorrect as there is only one decimal place.

Ignore Alphabetic Characters. If the expected answer is 10, and the student enters 10 apples, then only the numeric entry will be considered.

OBJECT FORMS

Line Standard Form: ax + by = c

Examples:

 x + 2y=3 true x +y - 3 =0 false

Note that you can include the "simplified" advanced option to ensure that the student's entry does not include fractional forms that are not simplified, e.g. x + 12/4 y = 3.

Line Slope Intercept Form: y = ax + b

Examples:

 y = 2x +3 true x +y - 3 =0 false

Line Point Slope Form: (y-b) = c(x-a)

Examples:

 (y - 3) = 2x true x +y - 3 =0 false

Quadratic Standard Form:  ax2 + bx + c = 0

Examples:

 x2 + 5x - 6 = 0 true x2 = 6 -5x false

Polynomial Standard Form: Σ (ai * xi) = 0  where ai are the coefficients and i is the degree of the polynomial)

Examples:

 x3 + x2 + 5x - 6 = 0 true x3 + x2 = 6 -5x false

Polynomial Factored Form: ∏(x-ai) = 0  e.g. (x-1)(x-2)

Conic Standard Form:

Circle: (x−h)2+(y−k)2=r2

Ellipse: (x−h)2/a2+(y−k)2/b2=1

Hyperbola: (x−h)2/a2-(y−k)2/b2=1

Parabola: 4p(y – k) = (x – h)2

Examples:

 x2 + (y-1)2 + = 4 circle true x2 + y2 -2y = 3 circle false x2 - (y-1)2 + = 4 hyperbola true x2 + ((y-1)/2)2 + = 4 ellipse true

Note that these conic forms are intended to have a direct geometric interpretation, so forms like these would not be evaluated correctly as the denominators do not conform to a direct geometric parameter.  Parabola Vertex Form: y = a(x – h)2 + k ( x or y can be on LHS)

 x = 2(y – 2)2 + 2 true y = x2 + 2 true x = y2 false