# Create Answer Keys for Questions Using Symbolic Mode

When typing the answer key for a question in Symbolic mode, you can include a broad range of mathematical expressions.

## Specifying Numeric Values in Answer Keys

Do not use commas to separate digits in numbers. This can cause ambiguous answer keys and let your students receive credit for incorrect answers. For example, x + 1,234 is interpreted both as 1234 + x and as a list with two elements: x + 1 and 234.

Do not use mixed numbers. Instead, use improper fractions or express the mixed number as a sum, for example 7/4 or 1+3/4.

## Specifying Math Variables in Answer Keys

Although not always required, you can declare your math variables at the beginning of the answer key in a comma-delimited list ending with a colon, as in the following example.

x_1, y_1, x_2, y_2: sqrt((x_1-x_2)^2 + (y_1-y_2)^2) 

Variable names in answer keys must observe the following conventions:

• Variable names must include only letters, numbers, and underscores; underscores in variable names indicate subscripting.
• Variable names are case-sensitive; X is different from x.
• Variable names should not duplicate the names of functions or of the numeric value pi.
• Variables having the names of Greek letters are displayed in mathPad and calcPad as the corresponding Greek letters. Greek letters except for π are treated as variables. The letter π is treated as a constant.
• The variable e can be used, but it will be treated as both Euler's number and as a variable name, so either response is marked correct.

The following table lists some examples.

Math Notation

x
x
x1
x1
x1
x_1
books
books
λ
lambda

Be sure that your question identifies any variables that the student should use in the answer.

## Specifying Perl Variables in Answer Keys

If you have defined a Perl variable for use in your answer key — for example, to randomize numeric values in your question — always enclose it with the <EQN> tag as in the following example:

<EQN $d>x + <EQN$e>

To avoid confusion, use different names for Perl variables and math variables in your question. For more information variables, see Perl Variables for Math Questions (Algebraic and Symbolic Modes) and Default Values and Tolerance for Symbolic Evaluation

## Specifying Math Expressions in Answer Keys

The functions and values in the following table are case-sensitive- for example, ABS(x) cannot be substituted for abs(x).

For many functions in the following table, parentheses can be omitted if the argument is simple and unambiguous — for example, a single variable or constant. Include parentheses when you need to ensure that a specific order of operations is observed. The default order of operations for symbolic answer keys is: subscripts, then factorials, then exponentiation, then multiplication and division, then addition and subtraction.

Expression

Math Notation

Notes

x + y
x + y

Subtraction

xy
x - y

Multiplication

2x
2 ∙ x
2 × x

2x
2 * x

No distinction is made between explicit or implicit multiplication.

Division or fractions

x ÷ 3
x / 3

No distinction is made between responses specified as stacked fractions or using the division sign (÷).

Exponentiation

x3
x^3
x**3

Square root

sqrt(x)

Other roots

rootn(x)
root(x,n)

rootn(x) works only when n is an integer.

Subscript

xn
x(a + b)

x_n
x_(a + b)

If the subscript includes mathematical operators, including implicit multiplication, enclose it in parentheses.

Factorial

x!
x!

Factorials are calculated only for natural numbers.

Absolute value

|x|
abs(x)

Greek letters

α + β
Ω

alpha + beta
Omega

Type the name of lowercase Greek letters in lowercase. Type the names of uppercase Greek letters in proper case.

Greek letters except for π are treated as variables. The letter π is treated as a constant.

In mathPad, your students must type the names of all Greek letters except for π and θ.

Pi (π

π
pi

Substituting 3.14 only approximates this value. You and your students should use pi to indicate the exact value of pi unless the question specifically instructs the student to use an approximation to pi.

Euler's number

e
e
exp(1)

Exponential function

e3
e^3
exp(3)

Logarithm (base 10)

log x
log10 x

log x
log(x)

Logarithm (arbitrary base)

log16(x)
logb(x)

log_16(x)
log_(b)(x)

If the base is anything other than a number, enclose it in parentheses.

Natural logarithm

ln x
ln(x)
ln x

Grouping, Order of Operations

4 (x + 1)
4 [x + 1]
4 {x + 1}

4(x + 1)
4[x + 1]
4{x + 1}

No distinction is made among the three types of grouping symbols.

Parentheses can also be used to delimit ordered tuples; braces can also be used to delimit unordered lists.

Scientific / "e" notation

2.46 × 106
2.46e+6

2.46 * 10^6
2.46e+6

Trigonometric functions

sin x
cos x
tan θ
cot (πθ)
sec A
csc x

sin x
cos(x)
tan theta
cot(pi - theta)
sec(A)
csc x

Inverse trigonometric functions

arcsin x
sin-1(x)
arccos x
cos-1(x)
arctan x
tan-1(x)
arccot x
cot-1(x)
arcsec x
sec-1(x)
arccsc x
csc-1(x)

arcsin x
sin^(-1)(x)
arccos x
cos^(-1)(x)
arctan x
tan^(-1)(x)
arccot x
cot^(-1)(x)
arcsec x
sec^(-1)(x)
arccsc x
csc^(-1)(x)

For each inverse trigonometric function, you can abbreviate "arc" to "a" as in asin(x).

Hyperbolic functions

sinh x
cosh x
tanh x
coth x
sech x
csch x

sinh x
cosh x
tanh x
coth x
sech x
csch x

Specify inverse hyperbolic functions using the -1 notation as for trigonometric functions.

Ordered pairs, ordered tuples

(x, y)
(x, y, z)

(x, y)
(x, y, z)

Comma-delimited lists in parentheses are evaluated as ordered tuples.

Sets, unordered lists of elements

{1, 2}
3, 4

{1, 2}
3, 4

To accept only standard roster notation with braces, set $ROSTER_ONLY=1. By default, your students' responses will match your key if they enumerate every element at least once, regardless of repetition. Thus, {1,2,2} = {1,2}. To require your students to enumerate all instances of repeated elements in a set, set $NO_REPETITION=1.

Infinity

infinity

Undefined

UNDEFINED
UNDEFINED

Degree

30°
30 deg

Degrees are not evaluated mathematically by default. For this reason, your students must enter the exact form of the answer that you provide and not a mathematically- equivalent response. For example, if you specified cos(60 deg), your students would be marked incorrect for submitting either 0.5 or sin(30°).

Imaginary unit

i
i

No solution

NO SOLUTION
NO SOLUTION

Empty set

empty
empty